Full article: Kantorovich-Stancu type (α,λ,s) - Bernstein operators and their approximation properties

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ABSTRACT

In this study, we establish a new class of Kantorovich-Stancu type $(α, λ, s) -$ Bernstein operators via an adaptation of Bézier bases which are formulated with the inclusion of the shape parameters $λ \in [- 1, 1]$ , $α \in [0, 1]$ , and a positive parameter $s .$ First, we present a uniform convergence result for these operators and, subsequently, examine the convergence properties by utilizing the weighted $B$ -statistical convergence notion. Furthermore, we estimate the rate of the weighted $B$ -statistical convergence of these operators. We conclude our work by providing a numerical example with explanatory graphs to show their approximation behaviours.

KEYWORDS:

1. Introduction

In the field of mathematical analysis, the Weierstrass Approximation Theorem (Weierstrass Citation1885) has immense significance since it states that any continuous function $ξ (τ)$ on a closed and bounded interval can be uniformly approximated on that interval by a sequence of polynomials $p_{μ} (τ)$ within a bounded error. Consequently, there have been different approaches to the statement and the proof of the Weierstrass Approximation Theorem. In 1912, one of the most famous advances to prove this landmark theorem was implemented by Sergei N. Bernstein (Citation1912) via the so-called Bernstein polynomials. In order to achieve his goal, Bernstein (Citation1912) defined the following Bernstein operators

(1)

B_{μ} (ξ; τ) = \sum_{κ = 0}^{μ} b_{μ, κ} (τ) ξ (\frac{κ}{μ}),

(1)

where $τ \in [0, 1],$ $μ \in N$ and

(2)

\begin{aligned} b_{μ, κ} (τ) = (\begin{matrix} μ \\ κ \end{matrix}) τ^{κ} {(1 - τ)}^{μ - κ}, \\ s . t . b_{μ, κ} (τ) = 0 if κ > μ or κ < 0, \end{aligned}

(2)

are the Bernstein basis functions. Bernstein-type operators have attracted great attention due to their invaluable traits listed below:

They have explicit forms and operate competently for the functions which are well-defined for rational-valued variables.
They possess numerous shape-preserving properties, such as linearity, monotonicity, endpoint interpolation, and convexity.
These operators, along with their variations, are easy to manipulate in computer programs and very convenient to work with if the structure of the function is intricate.
They have many crucial and impressive implementations in the fields of signal and image processes, solutions of differential and integral equations, robotics, and so on.
Even though they satisfy the Korovkin theorem in the classical sense, they also fulfill the different versions of Korovkin theorems which utilize the summability methods and statistical type convergences since they are well connected with the theory of infinite matrices and summability.

Because of these traits mentioned above, there has been tremendous research done regarding the approximation and shape-preserving properties, along with many other aspects of the Bernstein operators and their variations in the last century. Two of the most notable variations of classical Bernstein operators (1.1) were introduced by Kantorovich and Stancu in (Citation1930) and (Stancu Citation1968), respectively. The operators defined by Kantorovich, denoted as $K_{μ} (ξ; τ)$ , are an integral adaptation of operators in (1.1) and very convenient tools to approximate functions that are Riemann integrable on $[0, 1] .$ The operators symbolized as $S_{μ}^{σ, ρ} (ξ; τ),$ which are inaugurated by Stancu, involve two real parameters $σ$ and $ρ$ with $0 \leq ρ \leq σ .$ The purpose here was to give more flexibility in modelling with the inclusion of parameters $σ$ and $ρ .$ One can find more information on these operators and the subsequent studies in papers (Cai et al. Citation2022; Ayman Mursaleen et al. Citation2023, Citation2024; Bărbosu Citation2004; Bustamante Citation2017; Cai et al. Citation2018; Milovanović et al. Citation2018; Nasiruzzaman et al. Citation2019; Mohiuddine and Özger Citation2020; Özger Citation2019, Citation2020; Özger et al. Citation2020, Citation2022; Kadak and Özger Citation2021; Aktuğlu et al. Citation2022; Ansari et al. Citation2022a, Citation2022b; Aslan Citation2022, Citation2023, Citation2024; Bodur et al. Citation2022; Savaş and Mursaleen Citation2023; Srivastava et al. Citation2021) (and references therein), respectively.

Likewise, the Bernstein basis functions given in (2) have also been modified in order to achieve improved results on the surface and curve modellings. The one variation that intrigued us the most is called Bézier bases which were constructed by Ye et al. in (Ye et al. Citation2010) with the inclusion of shape parameter $λ \in [- 1, 1] .$ In a similar fashion, Chen et al. (Chen et al. Citation2017) defined the $α$ version of bases in (1.2) where $α \in [0, 1]$ , and subsequently, Gezer et al. (Citation2022) constructed the blending-type $(α, λ, s)$ -Bernstein operators by combining the shape parameters $λ,$ $α$ , and $s,$ where $s$ is a positive integer.

Gezer et al. defined the blending-type $(α, λ, s) -$ Bernstein operators by employing the sample values of $ξ$ on the interval $[0, 1] .$ Inspired by their work, we define the Kantorovich-Stancu type $(α, λ, s) -$ Bernstein operators ( $(α, λ, s)$ -BKSO) by taking into account the mean values of $ξ$ in the intervals $[\frac{κ + ρ}{μ + σ + 1}, \frac{κ + ρ + 1}{μ + σ + 1}]$ , where $0 \leq ρ \leq σ,$ instead of sample values of $ξ .$ In this paper, we aim to investigate the approximation and convergence properties of $(α, λ, s)$ -BKSO in the statistical sense. Moreover, we give an estimation of the rate of weighted $B$ -statistical convergence and prove a Voronovskaja-type approximation theorem by a family of linear operators using the notion of weighted $B$ -statistical convergence.

The manuscript’s organization is as follows: Section 2 summarizes the variations of Bernstein-type operators. Section 3 is devoted to the construction of $(α, λ, s)$ -BKSO and the auxiliary results regarding their moments, while section 4 provides our main results about the convergence properties of $(α, λ, s)$ -BKSO. Section 5 includes the explanatory figures, which show the approximation behaviour of $(α, λ, s)$ -BKSO and concluding remarks on our findings.

2. Essential preliminaries

In this section, we present the established work from the literature for the readers who are not familiar with the definitions of $α$ -Bernstein, $λ$ -Bernstein, blending $(α, λ, s)$ -Bernstein basis functions, and their corresponding operators. Firstly we note that the binomial coefficients are given by the formula

(\begin{matrix} μ \\ κ \end{matrix}) = \{\begin{matrix} \frac{μ!}{κ! (μ - κ)!}, 0 \leq κ \leq μ, \\ 0, otherwise . \end{matrix}

In (Chen et al. Citation2017), Chen et al. inaugurated a new type of operator, the so-called $α -$ Bernstein operator, in the literature, which is given by

(3)

T_{μ, α} (ξ; τ) = \sum_{κ = 0}^{μ} ψ_{μ, κ}^{(α)} (τ) ξ (\frac{κ}{μ}),

(3)

where real number $α$ is the shape parameter. For $μ \geq 2,$ $α, τ \in [0, 1],$ and $ξ (τ) \in C [0, 1],$ the $α -$ Bernstein basis is represented by $ψ_{1, 0}^{(α)} (τ) = 1 - τ$ , $ψ_{1, 1}^{(α)} (τ) = τ$ and

ψ_{μ, κ}^{(α)} (τ) = [(1 - α) (\begin{matrix} μ - 2 \\ κ \end{matrix}) τ + (1 - α) (\begin{matrix} μ - 2 \\ κ - 2 \end{matrix}) (1 - τ) + α (\begin{matrix} μ \\ κ \end{matrix}) τ (1 - τ)] τ^{κ - 1} (1 - τ)^{μ - κ - 1} .

If one chooses $α = 1$ , then (3) reduces to the classical Bernstein operators given by (1). This implies that the $α -$ Bernstein operator is more efficient than the classical Bernstein operator. There are a vast number of studies built on the $α -$ Bernstein operators in the literature (see (Mohiuddine and Özger Citation2020; Aktuğlu et al. Citation2022; Chen et al. Citation2022) and references therein).

A decade ago, Ye et al. constructed the new Bézier bases with shape parameter $λ \in [- 1, 1]$ by

(4)

{\tilde{b}}_{μ, κ} (λ; τ) = \{\begin{matrix} b_{μ, 0} (τ) - \frac{λ}{μ + 1} b_{μ + 1, 1} (τ), & κ = 0, \\ b_{μ, κ} (τ) + λ (\frac{μ - 2 κ + 1}{μ^{2} - 1} b_{μ + 1, κ} (τ) - \frac{μ - 2 κ - 1}{μ^{2} - 1} b_{μ + 1, κ + 1} (τ)), & κ = \overline{1, μ - 1}, \\ b_{μ, μ} (τ) - \frac{λ}{μ + 1} b_{μ + 1, μ} (τ), & κ = μ, \end{matrix}

(4)

where $b_{μ, κ} (τ)$ are defined as in (2) (see (Ye et al. Citation2010)). Later on, Cai et al. (Citation2018) established the corresponding $λ -$ Bernstein operators as

(5)

B_{μ, λ} (ξ; τ) = \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ} (λ; τ) ξ (\frac{κ}{μ}) .

(5)

Note that for $λ = 0,$ Bézier bases in (4) and $λ -$ Bernstein operators in (5) transform to the classical Bernstein bases and operators given by (2) and (1), respectively.

Next, we give the following lemma, which gives the moments of the $λ -$ Bernstein operators.

Lemma 1

([14]). For $λ -$ Bernstein operators, the following equalities hold true:

B_{μ, λ} (1; τ) = 1;

B_{μ, λ} (t; τ) = τ + λ [\frac{1 - 2 τ + τ^{μ + 1} - {(1 - τ)}^{μ + 1}}{μ (μ - 1)}];

B_{μ, λ} (t^{2}; τ) = τ^{2} + \frac{τ (1 - τ)}{μ} + λ [\frac{2 τ - 4 τ^{2} + 2 τ^{μ + 1}}{μ (μ - 1)} + \frac{τ^{μ + 1} + {(1 - τ)}^{μ + 1} - 1}{μ^{2} (μ - 1)}];

\begin{aligned} B_{μ, λ} (t^{3}; τ) = τ^{3} + \frac{3 τ^{2} (1 - τ)}{μ} + \frac{2 τ^{3} - 3 τ^{2} + τ}{μ^{2}} + λ [\frac{- 6 τ^{3} + 6 τ^{μ + 1}}{μ^{2}} + \frac{3 τ^{2} - 3 τ^{μ + 1}}{μ (μ - 1)} \break + \frac{- 9 τ^{2} + 9 τ^{μ + 1}}{μ^{2} (μ - 1)} + \frac{- 4 τ + 4 τ^{μ + 1}}{μ^{3} (μ - 1)} + \frac{(1 - τ^{μ + 1} - {(1 - τ)}^{μ + 1}) (μ + 3)}{μ^{3} (μ^{2} - 1)}]; \end{aligned}

\begin{aligned} B_{μ, λ} (t^{4}; τ) = τ^{4} + \frac{6 (τ^{3} - τ^{4})}{μ} + \frac{7 τ^{2} - 18 τ^{3} + 11 τ^{4}}{μ^{2}} + \frac{τ - 7 τ^{2} + 12 τ^{3} - 6 τ^{4}}{μ^{3}} \\ + λ [\frac{6 τ^{2} - 2 τ^{3} - 8 τ^{4} + 4 τ^{μ + 1}}{μ^{2}} + \frac{- τ^{2} - 32 τ^{3} + 16 τ^{4} + 17 τ^{μ + 1}}{μ^{3}} + \frac{τ - τ^{μ + 1}}{μ^{4}} \break + \frac{7 τ^{2} - 7 τ^{μ + 1}}{μ^{2} (μ - 1)} \\ + \frac{τ - 23 τ^{2} + 22 τ^{μ + 1}}{μ^{3} (μ - 1)} + \frac{{(1 - τ)}^{μ + 1} + τ - 1}{μ^{4} (μ - 1)}] . \end{aligned}

Recently, generalized blending-type $α$ -Bernstein operators, which utilize shape parameters $α$ and $s,$ are constructed by Aktuğlu et al. in (Citation2022) as

(6)

L_{μ}^{α, s} (ξ; τ) = \sum_{κ = 0}^{μ} [(1 - α) (\begin{matrix} μ - s \\ κ - s \end{matrix}) τ^{κ - s + 1} {(1 - τ)}^{μ - κ} \break + (1 - α) (\begin{matrix} μ - s \\ κ \end{matrix}) τ^{κ} {(1 - τ)}^{μ - s - κ + 1} + α (\begin{matrix} μ \\ κ \end{matrix}) τ^{κ} {(1 - τ)}^{μ - κ}] ξ (\frac{κ}{μ})

(6)

for $μ \geq s,$ and

(7)

L_{μ}^{α, s} (ξ; τ) = \sum_{κ = 0}^{μ} (\begin{matrix} μ \\ κ \end{matrix}) τ^{κ} {(1 - τ)}^{κ} ξ (\frac{κ}{μ}),

(7)

for $μ < s$ , where $α, τ \in [0, 1],$ $ξ (τ) \in C [0, 1]$ and $s \in Z^{+} = \{1, 2, \dots\}$ . The choices of $s = 1$ and $s = 2$ in operators (6) and (7) result in the classical Bernstein operators and $α -$ Bernstein operators given by (1) and (3), respectively.

Subsequently, in (Gezer et al. Citation2022) introduced the blending-type $(α, λ, s)$ -Bernstein operators to the literature as follows:

(8)

L_{μ, λ}^{(α, s)} (ξ; τ) = \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) ξ (\frac{κ}{μ}),

(8)

where $α \in [0, 1],$ $λ \in [- 1, 1],$ $s$ is a positive integer and

(9)

{\tilde{b}}_{μ, κ}^{α, s} (λ; τ) = \{\begin{matrix} {\tilde{b}}_{μ, κ} (λ; τ) & if μ < s, \\ (1 - α) [τ {\tilde{b}}_{μ - s, κ - s} (λ; τ) + (1 - τ) {\tilde{b}}_{μ - s, κ} (λ; τ)] + α {\tilde{b}}_{μ, κ} (λ; τ) & if μ \geq s, \end{matrix}

(9)

are the blending-type $(α, λ, s)$ -Bernstein bases with ${\tilde{b}}_{μ, κ} (λ; τ)$ defined as in (4). Note that $L_{μ, λ}^{(α, s)} (ξ; τ)$ reduces to the classical Bernstein operators if $α = 1$ and $λ = 0,$ the $α -$ Bernstein operators if $λ = 0$ and $s = 2,$ the $λ -$ Bernstein operators if $α = 1$ and $s = 1,$ and lastly the blending-type $α -$ Bernstein operators if $λ = 0$ .

For the sake of more precise and straightforward computations, Gezer et al. reformulated the blending-type $(α, λ, s)$ -Bernstein operators given in Equation (8) in the following manner.

Theorem 1

([23]). For any $α \in [0, 1],$ $λ \in [- 1, 1]$ and a positive integer $s$ ,

L_{μ, λ}^{(α, s)} (ξ; τ) = \{\begin{matrix} B_{μ, λ} (ξ; τ) & if μ < s, \\ (1 - α) B_{μ, λ}^{s, (⋆)} (ξ; τ) + α B_{μ, λ} (ξ; τ) & if μ \geq s, \end{matrix}

where

B_{μ, λ} (ξ; τ) = \sum_{κ = 0}^{μ} b_{μ, κ} (τ) ξ (\frac{κ}{μ}) + λ \sum_{κ = 0}^{μ - 1} \frac{μ - 2 κ - 1}{μ^{2} - 1} b_{μ + 1, κ + 1} (τ) [ξ (\frac{κ + 1}{μ}) - ξ (\frac{κ}{μ})],

and

B_{μ, λ}^{s, (⋆)} (ξ; τ) = \sum_{κ = 0}^{μ - s} b_{μ - s, κ} (τ) [τξ (\frac{κ + s}{μ}) + (1 - τ) ξ (\frac{κ}{μ})]

+ λτ \sum_{κ = 0}^{μ - s - 1} \frac{μ - s - 2 κ - 1}{{(μ - s)}^{2} - 1} b_{μ - s + 1, κ + 1} (τ) [ξ (\frac{κ + s + 1}{μ}) - ξ (\frac{κ + s}{μ})]

+ λ (1 - τ) \sum_{κ = 0}^{μ - s - 1} \frac{μ - s - 2 κ - 1}{{(μ - s)}^{2} - 1} b_{μ - s + 1, κ + 1} (τ) [ξ (\frac{κ + 1}{μ}) - ξ (\frac{κ}{μ})] .

Next, we give the following lemmas, which present the moments of the blending-type $(α, λ, s)$ -Bernstein operators.

(10)

L_{μ, λ}^{(α, s)} (1; τ) = 1;

(10)

Lemma 2

([23]). If $μ < s,$ for any $α \in [0, 1]$ and $λ \in [- 1, 1]$ we have $L_{μ, λ}^{(α, s)} (ξ; τ) = B_{μ, λ} (ξ; τ)$ where $B_{μ, λ} (ξ; τ)$ are defined as in equation (5) and have the moments given as in Lemma 1.

Lemma 3

([23]). If $μ \geq s$ , for any $α \in 0, 1]$ and $λ \in [- 1, 1]$ , we have the following identities:

(11)

L_{μ, λ}^{(α, s)} (1; τ) = 1;

(11)

L_{μ, λ}^{(α, s)} (t; τ) = τ + \frac{αλ}{μ} [\frac{1 - 2 τ + τ^{μ + 1} - {(1 - τ)}^{μ + 1}}{(μ - 1)}] + \frac{(1 - α) λ}{μ} [\frac{1 - 2 τ + τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}}{(μ - s - 1)}];

L_{μ, λ}^{(α, s)} (t^{2}; τ) = τ^{2} + \frac{[μ + (1 - α) s (s - 1)] τ (1 - τ)}{μ^{2}} + \frac{αλ}{μ} [\frac{2 τ - 4 τ^{2} + 2 τ^{μ + 1}}{(μ - 1)}] + \frac{αλ}{μ^{2}} [\frac{τ^{μ + 1} + {(1 - τ)}^{μ + 1} - 1}{(μ - 1)}]

+ \frac{(1 - α) λ}{μ} [\frac{2 τ - 4 τ^{2} + 2 τ^{μ - s + 1}}{(μ - s - 1)}] + \frac{(1 - α) λ}{μ^{2}} [\frac{τ^{μ - s + 1} + {(1 - τ)}^{μ - s + 1} - 1 + 2 sτ (τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1})}{(μ - s - 1)}] .

Remark 1. In (Gezer et al. Citation2022), Gezer et al. only presented and proved the moments given by (10), (11) and (12) for $μ \geq s .$ In the next lemma, we will complete the unfinished part of Lemma 3 by stating and giving the proofs of $L_{μ, λ}^{(α, s)} (t^{3}; τ)$ and $L_{μ, λ}^{(α, s)} (t^{4}; τ)$ .

Lemma 4.

If $μ \geq s$ , for any $α \in [0, 1]$ and $λ \in [- 1, 1]$ , we have the moments $L_{μ, λ}^{(α, s)} (t^{3}; τ)$ and $L_{μ, λ}^{(α, s)} (t^{4}; τ)$ as follows:

L_{μ, λ}^{(α, s)} (t^{3}; τ) = τ^{3} + \frac{3 [μ + (1 - α) s (s - 1)] τ^{2} (1 - τ)}{μ^{2}} + \frac{[μ + (1 - α) s (s^{2} - 1)] τ (1 - 3 τ + 2 τ^{2})}{μ^{3}}

+ αλ [\frac{- 6 τ^{3} + 6 τ^{μ + 1}}{μ^{2}} + \frac{3 τ^{2} - 3 τ^{μ + 1}}{μ (μ - 1)} + \frac{- 9 τ^{2} + 9 τ^{μ + 1}}{μ^{2} (μ - 1)} + \frac{- 4 τ + 4 τ^{μ + 1}}{μ^{3} (μ - 1)}

+ \frac{(1 - τ^{μ + 1} - {(1 - τ)}^{μ + 1}) (μ + 3)}{μ^{3} (μ^{2} - 1)}]

+ (1 - α) λ [\frac{- 6 τ^{3} + 6 τ^{μ - s + 1}}{μ^{2}} + \frac{3 τ^{2} - 3 τ^{μ - s + 1}}{μ (μ - s - 1)} + \frac{- 9 τ^{2} + 9 τ^{μ - s + 1}}{μ^{2} (μ - s - 1)} + \frac{- 4 τ + 4 τ^{μ - s + 1}}{μ^{3} (μ - s - 1)}

+ \frac{(1 - τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}) (μ - s + 3)}{μ^{3} ({(μ - s)}^{2} - 1)}]

+ \frac{(1 - α) λ}{μ^{2}} [\frac{6 s τ (- τ^{2} + τ^{μ - s + 1})}{(μ - s - 1)}] + \frac{(1 - α) λ}{μ^{3}} [\frac{3 s (s - 1) (τ - 3 τ^{2} + 2 τ^{3} + τ^{μ - s + 1})}{(μ - s - 1)}]

(13)

+ \frac{(1 - α) λ}{μ^{3}} [\frac{3 τs (s - 1) (- τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1})}{(μ - s - 1)}];

(13)

(14)

L_{μ, λ}^{(α, s)} (t^{4}; τ) = τ^{4} + \frac{6 [μ + (1 - α) s (s - 1)] τ^{3} (1 - τ)}{μ^{2}} + \frac{[μ + (1 - α) s (s^{2} - 1)] τ^{2} (7 - 18 τ + 11 τ^{2})}{μ^{3}}

(14)

+ \frac{[μ + (1 - α) s (s^{3} - 1)] τ (1 - 7 τ + 12 τ^{2} - 6 τ^{3})}{μ^{4}} + \frac{3 (1 - α) s {(s - 1)}^{2} (μ - s) τ^{2} (- 1 + 2 τ - τ^{2})}{μ^{4}}

+ αλ [\frac{6 τ^{2} - 2 τ^{3} - 8 τ^{4} + 4 τ^{μ + 1}}{μ^{2}} + \frac{- τ^{2} - 32 τ^{3} + 16 τ^{4} + 17 τ^{μ + 1}}{μ^{3}} + \frac{τ - τ^{μ + 1}}{μ^{4}}

+ \frac{7 τ^{2} - 7 τ^{μ + 1}}{μ^{2} (μ - 1)} + \frac{τ - 23 τ^{2} + 22 τ^{μ + 1}}{μ^{3} (μ - 1)} + \frac{{(1 - τ)}^{μ + 1} + τ - 1}{μ^{4} (μ - 1)}]

+ (1 - α) λ [\frac{6 τ^{2} - 2 τ^{3} - 8 τ^{4} + 4 τ^{μ - s + 1}}{μ^{2}} + \frac{- τ^{2} - 32 τ^{3} + 16 τ^{4} + 17 τ^{μ - s + 1}}{μ^{3}} + \frac{τ - τ^{μ - s + 1}}{μ^{4}}

+ \frac{7 τ^{2} - 7 τ^{μ - s + 1}}{μ^{2} (μ - s - 1)} + \frac{τ - 23 τ^{2} + 22 τ^{μ - s + 1}}{μ^{3} (μ - s - 1)} + \frac{{(1 - τ)}^{μ - s + 1} + τ - 1}{μ^{4} (μ - s - 1)}]

+ \frac{(1 - α) λ}{μ^{2}} [\frac{4 s (τ^{3} - 2 τ^{4} + (3 τ - 2) τ^{μ - s + 1})}{(μ - s - 1)}]

+ \frac{(1 - α) λ}{μ^{3}} [\frac{s ((12 s - 13) τ^{2} - 4 (9 s - 1) τ^{3} + 8 (3 s - 1) τ^{4} + (5 + 12 s - 12 τ (s - 1)) τ^{μ - s + 1})}{(μ - s - 1)}]

+ \frac{(1 - α) λ}{μ^{4}} [\frac{s ((4 s^{2} - 6 s + 3) τ - 2 (s - 1) (10 s + 7) τ^{2} + 32 (s^{2} - 1) τ^{3} - 16 (s^{2} - 1) τ^{4})}{(μ - s - 1)}

- \frac{s (4 s^{2} - 6 s + 5) τ^{μ - s + 1}}{(μ - s - 1)}]

+ \frac{(1 - α) λ}{μ^{4}} [\frac{2 s τ (2 s^{2} - 3 s + 2) (τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1})}{(μ - s - 1)}] .

Proof.

Recall that

L_{μ, λ}^{(α, s)} (ξ; τ) = (1 - α) B_{μ, λ}^{s, (⋆)} (ξ; τ) + α B_{μ, λ} (ξ; τ)

by Theorem 1 when $μ \geq s$ . Hence, it is sufficient to calculate $B_{μ, λ}^{s, (⋆)} (t^{3}; τ)$ , since $B_{μ, λ} (t^{3}; τ)$ is already given in Lemma 1. Therefore, for $ξ (t) = t^{3},$ we have

B_{μ, λ}^{s, (⋆)} (t^{3}; τ) = \sum_{κ = 0}^{μ - s} b_{μ - s, κ} (τ) [τ {(\frac{κ + s}{μ})}^{3} + (1 - τ) {(\frac{κ}{μ})}^{3}] + λτ \sum_{κ = 0}^{μ - s - 1} \frac{μ - s - 2 κ - 1}{{(μ - s)}^{2} - 1} b_{μ - s + 1, κ + 1} (τ) [{(\frac{κ + s + 1}{μ})}^{3} - {(\frac{κ + s}{μ})}^{3}]

+ λ (1 - τ) \sum_{κ = 0}^{μ - s - 1} \frac{μ - s - 2 κ - 1}{{(μ - s)}^{2} - 1} b_{μ - s + 1, κ + 1} (τ) [{(\frac{κ + 1}{μ})}^{3} - {(\frac{κ}{μ})}^{3}]

= \sum_{κ = 0}^{μ - s} b_{μ - s, κ} (τ) [\frac{3 τ κ^{2} s}{μ^{3}} + \frac{3 τ κ s^{2}}{μ^{3}} + \frac{τ s^{3}}{μ^{3}} + \frac{κ^{3}}{μ^{3}}]

+ λ \sum_{κ = 0}^{μ - s - 1} \frac{μ - s - 2 κ - 1}{{(μ - s)}^{2} - 1} b_{μ - s + 1, κ + 1} (τ) [\frac{3 κ^{2}}{μ^{3}} + \frac{(6 τs + 3) κ}{μ^{3}} + \frac{(3 τ s^{2} + 3 τs + 1)}{μ^{3}}]

= τ^{3} + \frac{3 (s (s - 1) + μ) τ^{2} (1 - τ)}{μ^{2}} + \frac{(s (s^{2} - 1) + μ) τ (2 τ^{2} - 3 τ + 1)}{μ^{3}} + \frac{3 λ}{μ^{3} (μ - s + 1)} Δ_{3} (μ, s; τ)

+ \frac{(6 τs + 3) λ}{μ^{3} (μ - s + 1)} Δ_{2} (μ, s; τ) + \frac{(3 τ s^{2} + 3 τs + 1) λ}{μ^{3} (μ - s + 1)} Δ_{1} (μ, s; τ) + \frac{- 6 λ}{μ^{3} ({(μ - s)}^{2} - 1)} Δ_{4} (μ, s; τ)

+ \frac{(- 12 τs - 6) λ}{μ^{3} ({(μ - s)}^{2} - 1)} Δ_{3} (μ, s; τ) + \frac{(- 6 τ s^{2} - 6 τs - 2) λ}{μ^{3} ({(μ - s)}^{2} - 1)} Δ_{2} (μ, s; τ),

where

Δ_{1} (μ, s; τ) = \sum_{κ = 0}^{μ - s - 1} b_{μ - s + 1, κ + 1} (τ) = 1 - τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1},

Δ_{2} (μ, s; τ) = \sum_{κ = 0}^{μ - s - 1} κ b_{μ - s + 1, κ + 1} (τ) = (μ - s + 1) τ (1 - τ^{μ - s}) - Δ_{1} (μ, s; τ),

Δ_{3} (μ, s; τ) = \sum_{κ = 0}^{μ - s - 1} κ^{2} b_{μ - s + 1, κ + 1} (τ)

= (μ - s + 1) (μ - s) τ^{2} (1 - τ^{μ - s - 1}) + (μ - s + 1) τ (1 - τ^{μ - s}) - 2 Δ_{2} (μ, s; τ) - Δ_{1} (μ, s; τ),

(see (Gezer et al. Citation2022)), and

Δ_{4} (μ, s; τ) = \sum_{κ = 0}^{μ - s - 1} κ^{3} b_{μ - s + 1, κ + 1} (τ)

= ({(μ - s)}^{2} - 1) (μ - s) τ^{3} (1 - τ^{μ - s - 2}) + (μ - s + 1) (μ - s) τ^{2} (1 - τ^{μ - s - 1}) + (μ - s + 1) τ (1 - τ^{μ - s}) - Δ_{3} (μ, s; τ) - Δ_{2} (μ, s; τ) - Δ_{1} (μ, s; τ) .

By substitution of $Δ_{1}, Δ_{2}, Δ_{3}$ and $Δ_{4}$ in $B_{μ, λ}^{s, (⋆)} (t^{3}; τ),$ we get

(15)

B_{μ, λ}^{s, (⋆)} (t^{3}; τ) = τ^{3} + \frac{3 (s (s - 1) + μ) τ^{2} (1 - τ)}{μ^{2}} + + \frac{(s (s^{2} - 1) + μ) τ (2 τ^{2} - 3 τ + 1)}{μ^{3}}

(15)

+ λ [\frac{- 6 τ^{3} + 6 τ^{μ - s + 1}}{μ^{2}} + \frac{3 τ^{2} - 3 τ^{μ - s + 1}}{μ (μ - s - 1)} + \frac{- 9 τ^{2} + 9 τ^{μ - s + 1}}{μ^{2} (μ - s - 1)} + \frac{- 4 τ + 4 τ^{μ - s + 1}}{μ^{3} (μ - s - 1)}

+ \frac{(1 - τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}) (μ - s + 3)}{μ^{3} ({(μ - s)}^{2} - 1)}]

+ \frac{λ}{μ^{2}} [\frac{6 s τ (- τ^{2} + τ^{μ - s + 1})}{(μ - s - 1)}] + \frac{λ}{μ^{3}} [\frac{3 s (s - 1) (τ - 3 τ^{2} + 2 τ^{3} + τ^{μ - s + 1})}{(μ - s - 1)}]

+ \frac{λ}{μ^{3}} [\frac{3 sτ (s - 1) (- τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1})}{(μ - s - 1)}] .

Hence, by equations (2.13), (2.14) and Lemma 1, we obtain $L_{μ, λ}^{(α, s)} (t^{3}; τ)$ as in equation (2.11).

Next, we will compute $L_{μ, λ}^{(α, s)} (t^{4}; τ) .$ First we need to obtain $B_{μ, λ}^{s, (⋆)} (t^{4}; τ) .$ For $ξ (t) = t^{4},$ we have

B_{μ, λ}^{s, (⋆)} (t^{4}; τ) = \sum_{κ = 0}^{μ - s} b_{μ - s, κ} (τ) [τ {(\frac{κ + s}{μ})}^{4} + (1 - τ) {(\frac{κ}{μ})}^{4}] + λτ \sum_{κ = 0}^{μ - s - 1} \frac{μ - s - 2 κ - 1}{{(μ - s)}^{2} - 1} b_{μ - s + 1, κ + 1} (τ) [{(\frac{κ + s + 1}{μ})}^{4} - {(\frac{κ + s}{μ})}^{4}]

+ λ (1 - τ) \sum_{κ = 0}^{μ - s - 1} \frac{μ - s - 2 κ - 1}{{(μ - s)}^{2} - 1} b_{μ - s + 1, κ + 1} (τ) [{(\frac{κ + 1}{μ})}^{4} - {(\frac{κ}{μ})}^{4}]

= \sum_{κ = 0}^{μ - s} b_{μ - s, κ} (τ) [\frac{4 τ κ^{3} s}{μ^{4}} + \frac{6 τ κ^{2} s^{2}}{μ^{4}} + \frac{4 τ κ s^{3}}{μ^{4}} + \frac{τ s^{4}}{μ^{4}} + \frac{κ^{4}}{μ^{4}}]

+ λ \sum_{κ = 0}^{μ - s - 1} \frac{μ - s - 2 κ - 1}{{(μ - s)}^{2} - 1} b_{μ - s + 1, κ + 1} (τ) [\frac{4 κ^{3}}{μ^{4}} + \frac{(12 τs + 6) κ^{2}}{μ^{4}} + \frac{(12 τ s^{2} + 12 τs + 4) κ}{μ^{4}} \break + \frac{(4 τ s^{3} + 6 τ s^{2} + 4 τs + 1)}{μ^{4}}]

= τ^{4} + \frac{6 (s (s - 1) + μ) τ^{3} (- τ + 1)}{μ^{2}} + \frac{(s (s^{2} - 1) + μ) τ^{2} (11 τ^{2} - 18 τ + 7)}{μ^{3}}

+ \frac{(s (s^{3} - 1) + μ) τ (- 6 τ^{3} + 12 τ^{2} - 7 τ + 1)}{μ^{4}} + \frac{3 s {(s - 1)}^{2} (μ - s) τ^{2} (- τ^{2} + 2 τ - 1)}{μ^{4}}

+ \frac{4 λ}{μ^{4} (μ - s + 1)} Δ_{4} (μ, s; τ) + \frac{(12 τs + 6) λ}{μ^{4} (μ - s + 1)} Δ_{3} (μ, s; τ) + \frac{(12 τ s^{2} + 12 τs + 4) λ}{μ^{4} (μ - s + 1)} Δ_{2} (μ, s; τ)

+ \frac{(4 τ s^{3} + 6 τ s^{2} + 4 τs + 1) λ}{μ^{4} (μ - s + 1)} Δ_{1} (μ, s; τ) + \frac{- 8 λ}{μ^{4} ({(μ - s)}^{2} - 1)} Δ_{5} (μ, s; τ) + \frac{(- 24 τs - 12) λ}{μ^{4} ({(μ - s)}^{2} - 1)} Δ_{4} (μ, s; τ)

+ \frac{(- 24 τ s^{2} - 24 τs - 8) λ}{μ^{4} ({(μ - s)}^{2} - 1)} Δ_{3} (μ, s; τ) + \frac{(- 8 τ s^{3} - 12 τ s^{2} - 8 τs - 2) λ}{μ^{4} ({(μ - s)}^{2} - 1)} Δ_{2} (μ, s; τ),

where

Δ_{5} (μ, s; τ) = \sum_{κ = 0}^{μ - s - 1} κ^{4} b_{μ - s + 1, κ + 1} (τ)

= ({(μ - s)}^{2} - 1) (μ - s) (μ - s - 2) τ^{4} (1 - τ^{μ - s - 3})

+ ({(μ - s)}^{2} - 1) (μ - s) τ^{3} (1 - τ^{μ - s - 2}) + (μ - s + 1) (μ - s) τ^{2} (1 - τ^{μ - s - 1})

+ (μ - s + 1) τ (1 - τ^{μ - s}) + Δ_{4} (μ, s; τ) - 3 Δ_{2} (μ, s; τ) - Δ_{1} (μ, s; τ) .

If we substitute $Δ_{1}, Δ_{2}, Δ_{3}, Δ_{4}$ , and $Δ_{5}$ in $B_{μ, λ}^{s, (⋆)} (t^{4}; τ),$ we obtain

(16)

B_{μ, λ}^{s, (⋆)} (t^{4}; τ) = τ^{4} + \frac{6 (s (s - 1) + μ) τ^{3} (- τ + 1)}{μ^{2}} + \frac{(s (s^{2} - 1) + μ) τ^{2} (11 τ^{2} - 18 τ + 7)}{μ^{3}}

(16)

+ \frac{(s (s^{3} - 1) + μ) τ (- 6 τ^{3} + 12 τ^{2} - 7 τ + 1)}{μ^{4}} + \frac{3 s {(s - 1)}^{2} (μ - s) τ^{2} (- τ^{2} + 2 τ - 1)}{μ^{4}}

+ λ [\frac{6 τ^{2} - 2 τ^{3} - 8 τ^{4} + 4 τ^{μ - s + 1}}{μ^{2}} + \frac{- τ^{2} - 32 τ^{3} + 16 τ^{4} + 17 τ^{μ - s + 1}}{μ^{3}} + \frac{τ - τ^{μ - s + 1}}{μ^{4}}

+ \frac{7 τ^{2} - 7 τ^{μ - s + 1}}{μ^{2} (μ - s - 1)} + \frac{τ - 23 τ^{2} + 22 τ^{μ - s + 1}}{μ^{3} (μ - s - 1)} + \frac{{(1 - τ)}^{μ - s + 1} + τ - 1}{μ^{4} (μ - s - 1)}]

+ \frac{λ}{μ^{2}} [\frac{4 s (τ^{3} - 2 τ^{4} + (3 τ - 2) τ^{μ - s + 1})}{(μ - s - 1)}]

+ \frac{λ}{μ^{3}} [\frac{s ((12 s - 13) τ^{2} - 4 (9 s - 1) τ^{3} + 8 (3 s - 1) τ^{4} + (5 + 12 s - 12 τ (s - 1)) τ^{μ - s + 1})}{(μ - s - 1)}]

+ \frac{λ}{μ^{4}} [\frac{s ((4 s^{2} - 6 s + 3) τ - 2 (s - 1) (10 s + 7) τ^{2} + 32 (s^{2} - 1) τ^{3})}{(μ - s - 1)}

+ \frac{s (- 16 (s^{2} - 1) τ^{4} - (4 s^{2} - 6 s + 5) τ^{μ - s + 1})}{(μ - s - 1)}]

+ \frac{λ}{μ^{4}} [\frac{2 s τ (2 s^{2} - 3 s + 2) (τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1})}{(μ - s - 1)}] .

Hence, by equations (2.13), (2.15) and Lemma 1, we attain $L_{μ, λ}^{(α, s)} (t^{4}; τ)$ as in equation (2.12). This completes the proof.

3. New (α, λ, s) – Bernstein-Kantorovich-Stancu operators of blending-type

This section is dedicated to the establishment and the associated auxiliary results of the operators $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ) .$ Let $L_{1} [0, 1]$ denote the space of all Lebesgue integrable functions on the interval $[0, 1]$ . Considering two non-negative parameters $σ$ and $ρ$ such that $0 \leq ρ \leq σ,$ we construct the positive $(α, λ, s)$ -BKSO as

(17)

S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ) = (μ + σ + 1) \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) \int_{\frac{κ + ρ}{μ + σ + 1}}^{\frac{κ + ρ + 1}{μ + σ + 1}} ξ (y) dy,

(17)

where $λ$ and $α$ are shape parameters, $s \in Z^{+}$ and ${\tilde{b}}_{μ, κ}^{α, s} (λ; τ)$ are as in (9).

Remark 2.

Recall that for $μ < s$ , the blending-type $(α, λ, s)$ -Bernstein bases are equal to Bézier bases, that is, ${\tilde{b}}_{μ, κ}^{α, s} (λ; τ) = {\tilde{b}}_{μ, κ} (λ; τ)$ by identity (9). Therefore, $(α, λ, s)$ -BKSO given by (17) transform to $λ$ -Bernstein-Kantorovich-Stancu operators $S_{μ, σ, ρ}^{(λ)} (ξ; τ)$ introduced by Bodur et al. in (Citation2022). They gave a uniform convergence result along with the order of approximation in the sense of local approximation and Voronovskaja type theorem with a complete proof for operators $S_{μ, σ, ρ}^{(λ)} (ξ; τ)$ . Even though the moments for the case $μ < s$ were presented in (Bodur et al. Citation2022), we provide the proofs for these identities for the readers’ convenience.

Lemma 5.

Let $α \in [0, 1],$ $λ \in [- 1, 1]$ , and $s \in Z^{+}$ . For $μ < s,$ the moments of $(α, λ, s)$ -BKSO are as follows:

(18)

S_{μ, σ, ρ}^{(α, λ, s)} (1; τ) = 1;

(18)

(19)

S_{μ, σ, ρ}^{(α, λ, s)} (t; τ) = \frac{2 μτ + 2 ρ + 1}{2 (μ + σ + 1)} + \frac{λ}{(μ - 1) (μ + σ + 1)} [1 - 2 τ + τ^{μ + 1} - {(1 - τ)}^{μ + 1}];

(19)

(20)

S_{μ, σ, ρ}^{(α, λ, s)} (t^{2}; τ) = \frac{(μ^{2} - μ) τ^{2} + 2 μ (1 + ρ) τ}{{(μ + σ + 1)}^{2}} + \frac{1 + 3 ρ + 3 ρ^{2}}{3 (μ + σ + {1)}^{2}}

(20)

+ \frac{2 λ}{(μ - 1) {(μ + σ + 1)}^{2}} [ρ + (μ - 2 ρ - 1) τ - 2 μ τ^{2} + (μ + ρ + 1) τ^{μ + 1} - ρ {(1 - τ)}^{μ + 1}];

(21)

S_{μ, σ, ρ}^{(α, λ, s)} (t^{3}; τ) = \frac{(μ^{3} - 3 μ^{2} + 2 μ) τ^{3} + (3 μ^{2} - 3 μ) (1 + ρ) τ^{2} + (2 + 6 ρ + 3 ρ^{2}) μτ}{{(μ + σ + 1)}^{3}}

(21)

+ \frac{(3 μ^{2} - 3 μ) τ^{2} + 3 μτ}{2 (μ + σ + {1)}^{3}} + \frac{(1 + 4 ρ + 6 ρ^{2} + 4 ρ^{3})}{4 (μ + σ + {1)}^{3}} + \frac{6 μλ}{{(μ + σ + 1)}^{3}} [- τ^{3} + τ^{μ + 1}]

+ \frac{λ}{(μ - 1) {(μ + σ + 1)}^{3}} \{1 + 3 ρ^{2} + 3 [μ (1 + 2 ρ) - 2 - 2 ρ - 2 ρ^{2}] τ + 3 [μ^{2} - μ (5 + 4 ρ)] τ^{2}

+ [- 3 μ^{2} + 6 μ (2 + ρ) + 5 + 6 ρ + 3 ρ^{2}] τ^{μ + 1} - [1 + 3 ρ^{2}] {(1 - τ)}^{μ + 1}\}

+ \frac{3 λ}{2 (μ - 1) {(μ + σ + 1)}^{3}} [τ^{μ + 1} + {(1 - τ)}^{μ + 1} - 1] + \frac{(μ + 3) λ}{(μ^{2} - 1) {(μ + σ + 1)}^{3}} [1 - τ^{μ + 1} - {(1 - τ)}^{μ + 1}];

(22)

S_{μ, σ, ρ}^{(α, λ, s)} (t^{4}; τ) = \frac{(μ^{4} - 6 μ^{3} + 11 μ^{2} - 6 μ) τ^{4} + (4 μ^{3} - 12 μ^{2} + 8 μ) (2 + ρ) τ^{3}}{{(μ + σ + 1)}^{4}}

(22)

+ \frac{(5 + 6 ρ + 2 ρ^{2}) (3 μ^{2} - 3 μ) τ^{2} + 2 (3 + 7 ρ + 6 ρ^{2} + 2 ρ^{3}) μτ}{{(μ + σ + 1)}^{4}} + \frac{(1 + 5 ρ + 10 ρ^{2} + 10 ρ^{3} + 5 ρ^{4})}{5 (μ + σ + {1)}^{4}}

+ \frac{λ}{{(μ + σ + 1)}^{4}} \{τ + [6 μ^{2} - μ] τ^{2} - 2 [μ^{2} + 2 μ (11 + 6 ρ)] τ^{3} - 8 [μ^{2} - 2 μ] τ^{4}\}

+ \frac{λ}{(μ - 1) {(μ + σ + 1)}^{4}} \{2 [1 + ρ - 2 ρ^{3}] (- 1 + {(1 - τ)}^{μ + 1})

\begin{aligned} + [μ (5 + 12 ρ + 12 ρ^{2}) - 9 - 24 ρ - 12 ρ^{2} - 8 ρ^{3}] τ \\ + [μ^{2} (13 + 12 ρ) - μ (49 + 60 ρ + 24 ρ^{2})] τ^{2} \end{aligned}

+ [- μ^{2} (13 + 12 ρ) + 4 μ (11 + 12 ρ + 3 ρ^{2}) + 11 + 26 ρ + 12 ρ^{2} + 4 ρ^{3}] τ^{μ + 1}\} .

Proof.

By utilizing the definitions of the operators $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ and $L_{μ, λ}^{(α, s)} (ξ; τ)$ given in (17) and (8), respectively, we have the first three moments of $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ as

S_{μ, σ, ρ}^{(α, λ, s)} (1; τ) = (μ + σ + 1) \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) \int_{\frac{κ + ρ}{μ + σ + 1}}^{\frac{κ + ρ + 1}{μ + σ + 1}} dt = (μ + σ + 1) \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) \frac{1}{(μ + σ + 1)}

= \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) = L_{μ, λ}^{(α, s)} (1; τ),

S_{μ, σ, ρ}^{(α, λ, s)} (t; τ) = (μ + σ + 1) \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) \int_{\frac{κ + ρ}{μ + σ + 1}}^{\frac{κ + ρ + 1}{μ + σ + 1}} tdt = (μ + σ + 1) \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) \frac{2 κ + 2 ρ + 1}{2 (μ + σ + {1)}^{2}}

= \frac{μ}{(μ + σ + 1)} L_{μ, λ}^{(α, s)} (t; τ) + \frac{2 ρ + 1}{2 (μ + σ + 1)} L_{μ, λ}^{(α, s)} (1; τ),

and

S_{μ, σ, ρ}^{(α, λ, s)} (t^{2}; τ) = (μ + σ + 1) \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) \int_{\frac{κ + ρ}{μ + σ + 1}}^{\frac{κ + ρ + 1}{μ + σ + 1}} t^{2} dt = (μ + σ + 1) \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) \frac{3 κ^{2} + (3 + 6 ρ) κ + 1 + 3 ρ + 3 ρ^{2}}{3 (μ + σ + {1)}^{3}}

= \frac{μ^{2}}{{(μ + σ + 1)}^{2}} L_{μ, λ}^{(α, s)} (t^{2}; τ) + \frac{(2 ρ + 1) μ}{{(μ + σ + 1)}^{2}} L_{μ, λ}^{(α, s)} (t; τ) + \frac{1 + 3 ρ + 3 ρ^{2}}{3 (μ + σ + {1)}^{2}} L_{μ, λ}^{(α, s)} (1; τ) .

Recall that for $μ < s,$ we have $L_{μ, λ}^{(α, s)} (ξ; τ) = B_{μ, λ} (ξ; τ)$ by Lemma 2. Bearing in mind this fact, we obtain the first three moments of $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ given in equations (3.2), (3.3) and (3.4) as

S_{μ, σ, ρ}^{(α, λ, s)} (1; τ) = B_{μ, λ} (1; τ) = 1,

S_{μ, σ, ρ}^{(α, λ, s)} (t; τ) = \frac{μ}{(μ + σ + 1)} B_{μ, λ} (t; τ) + \frac{2 ρ + 1}{2 (μ + σ + 1)} B_{μ, λ} (1; τ)

= \frac{2 μτ + 2 ρ + 1}{2 (μ + σ + 1)} + \frac{λ}{(μ - 1) (μ + σ + 1)} [1 - 2 τ + τ^{μ + 1} - {(1 - τ)}^{μ + 1}],

S_{μ, σ, ρ}^{(α, λ, s)} (t^{2}; τ) = \frac{μ^{2}}{{(μ + σ + 1)}^{2}} B_{μ, λ} (t^{2}; τ) + \frac{(2 ρ + 1) μ}{{(μ + σ + 1)}^{2}} B_{μ, λ} (t; τ) + \frac{1 + 3 ρ + 3 ρ^{2}}{3 (μ + σ + {1)}^{2}} B_{μ, λ} (1; τ)

= \frac{(μ^{2} - μ) τ^{2} + 2 μ (1 + ρ) τ}{{(μ + σ + 1)}^{2}} + \frac{1 + 3 ρ + 3 ρ^{2}}{3 (μ + σ + {1)}^{2}}

+ \frac{2 λ}{(μ - 1) {(μ + σ + 1)}^{2}} [ρ + (μ - 2 ρ - 1) τ - 2 μ τ^{2} + (μ + ρ + 1) τ^{μ + 1} - ρ {(1 - τ)}^{μ + 1}],

respectively. The proofs of equations (3.5) and (3.6) are omitted due to the similarity to the above calculations. Hence, the proof is complete.

Lemma 6.

Let $α \in [0, 1],$ $λ \in [- 1, 1]$ , and $s \in Z^{+}$ . For $μ \geq s,$ the moments of $(α, λ, s)$ -BKSO are as follows:

(23)

S_{μ, σ, ρ}^{(α, λ, s)} (1; τ) = 1;

(23)

(24)

S_{μ, σ, ρ}^{(α, λ, s)} (t; τ) = \frac{2 μτ + 2 ρ + 1}{2 (μ + σ + 1)} + \frac{αλ}{(μ - 1) (μ + σ + 1)} [1 - 2 τ + τ^{μ + 1} - {(1 - τ)}^{μ + 1}] + \frac{(1 - α) λ}{(μ - s - 1) (μ + σ + 1)} [1 - 2 τ + τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}]

(24)

S_{μ, σ, ρ}^{(α, λ, s)} (t^{2}; τ) = \frac{μ^{2} τ^{2} + (1 + 2 ρ) μτ + [μ + (1 - α) s (s - 1)] τ (1 - τ)}{{(μ + σ + 1)}^{2}} + \frac{1 + 3 ρ + 3 ρ^{2}}{3 (μ + σ + {1)}^{2}}

+ \frac{2 αλ}{(μ - 1) {(μ + σ + 1)}^{2}} [ρ + (μ - 2 ρ - 1) τ - 2 μ τ^{2} + (μ + ρ + 1) τ^{μ + 1} - ρ {(1 - τ)}^{μ + 1}]

+ \frac{2 (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{2}} [ρ + (μ - 2 ρ - 1) τ - 2 μ τ^{2} + (μ + ρ + 1 + sτ) τ^{μ - s + 1} \break - (ρ + sτ) {(1 - τ)}^{μ - s + 1}];

(27)

+ \frac{2 (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{2}} [ρ + (μ - 2 ρ - 1) τ - 2 μ τ^{2} + (μ + ρ + 1 + sτ) τ^{μ - s + 1} \break - (ρ + sτ) {(1 - τ)}^{μ - s + 1}];

(27)

S_{μ, σ, ρ}^{(α, λ, s)} (t^{3}; τ) = \frac{μ^{3} τ^{3} + (1 + 3 ρ + 3 ρ^{2}) μτ}{{(μ + σ + 1)}^{3}} + \frac{(3 + 6 ρ) μ^{2} τ^{2}}{2 (μ + σ + {1)}^{3}} + \frac{1 + 4 ρ + 6 ρ^{2} + 4 ρ^{3}}{4 (μ + σ + {1)}^{3}}

+ \frac{3 μ [μ + (1 - α) s (s - 1)] τ^{2} (1 - τ)}{{(μ + σ + 1)}^{3}} + \frac{[μ + (1 - α) s (s^{2} - 1)] τ (1 - 3 τ + 2 τ^{2})}{{(μ + σ + 1)}^{3}}

+ \frac{(3 + 6 ρ) [μ + (1 - α) s (s - 1)] τ (1 - τ)}{2 (μ + σ + {1)}^{3}} + \frac{6 μλ}{{(μ + σ + 1)}^{3}} [- τ^{3} + α τ^{μ + 1} + (1 - α) τ^{μ - s + 1}]

+ \frac{3 αλ}{(μ - 1) {(μ + σ + 1)}^{3}} \{ρ^{2} + [μ (1 + 2 ρ) - 2 - 2 ρ - 2 ρ^{2}] τ + [μ^{2} - μ (5 + 4 ρ)] τ^{2}

+ [- μ^{2} + 2 μ (2 + ρ) + 2 ρ + ρ^{2}] τ^{μ + 1} - ρ^{2} {(1 - τ)}^{μ + 1}\}

+ \frac{αλ}{2 (μ - 1) {(μ + σ + 1)}^{3}} [- 1 + 13 τ^{μ + 1} + {(1 - τ)}^{μ + 1}] + \frac{(μ + 3) α λ}{(μ^{2} - 1) {(μ + σ + 1)}^{3}} [1 - τ^{μ + 1} - {(1 - τ)}^{μ + 1}]

+ \frac{3 (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{3}} \{ρ^{2} + [μ (1 + 2 ρ) - 2 - 2 ρ - 2 ρ^{2}] τ + [μ^{2} - μ (5 + 4 ρ)] τ^{2}

- 2 μs τ^{3} - ρ^{2} {(1 - τ)}^{μ - s + 1} + [- μ^{2} + 2 μ (2 + ρ) + 2 ρ + ρ^{2}] τ^{μ - s + 1}\}

+ \frac{(1 - α) λ}{2 (μ - s - 1) {(μ + σ + 1)}^{3}} [- 1 + 13 τ^{μ - s + 1} + {(1 - τ)}^{μ - s + 1}]

+ \frac{3 s (s - 1) (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{3}} [τ - 3 τ^{2} + 2 τ^{3} + τ^{μ - s + 1}]

+ \frac{3 sτ (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{3}} [(2 μ - s + 2 ρ + 2) τ^{μ - s + 1} - (s + 2 ρ) {(1 - τ)}^{μ - s + 1}]

+ \frac{(μ - s + 3) (1 - α) λ}{({(μ - s)}^{2} - 1) {(μ + σ + 1)}^{3}} [1 - τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}];

S_{μ, σ, ρ}^{(α, λ, s)} (t^{4}; τ) = \frac{μ^{4} τ^{4} + (2 + 4 ρ) μ^{3} τ^{3} + (2 + 6 ρ + 6 ρ^{2}) μ^{2} τ^{2} + (1 + 4 ρ + 6 ρ^{2} + 4 ρ^{3}) μτ}{{(μ + σ + 1)}^{4}}

+ \frac{1 + 5 ρ + 10 ρ^{2} + 10 ρ^{3} + 5 ρ^{4}}{5 (μ + σ + {1)}^{4}} + \frac{3 (1 - α) s {(s - 1)}^{2} (μ - s) τ^{2} (- 1 + 2 τ - τ^{2})}{{(μ + σ + 1)}^{4}}

+ \frac{[μ + (1 - α) s (s^{3} - 1)]}{{(μ + σ + 1)}^{4}} [1 - 7 τ + 12 τ^{2} - 6 τ^{3}]

+ \frac{[μ + (1 - α) s (s^{2} - 1)] τ}{{(μ + σ + 1)}^{4}} [2 + 4 ρ + (7 μ - 12 ρ - 6) τ + 2 (- 9 μ + 4 ρ + 2) τ^{2} + 11 μ τ^{3}]

+ \frac{2 [μ + (1 - α) s (s - 1)] τ}{{(μ + σ + 1)}^{4}} \{1 + 3 ρ + 3 ρ^{2} + [3 μ (1 + 2 ρ) - 1 - 3 ρ - 3 ρ^{2}] τ

+ 3 [μ^{2} - μ (1 + 2 ρ)] τ^{2} - 3 μ^{2} τ^{3}\}

+ \frac{λ}{{(μ + σ + 1)}^{4}} \{τ + [6 μ^{2} - μ] τ^{2} - 2 [μ^{2} + 2 μ (11 + 6 ρ)] τ^{3} - 8 [μ^{2} - 2 μ] τ^{4}\}

+ \frac{(4 μ^{2} + μ (29 + 24 ρ) - 1) λ}{{(μ + σ + 1)}^{4}} [α τ^{μ + 1} + (1 - α) τ^{μ - s + 1}]

+ \frac{αλ}{(μ - 1) {(μ + σ + 1)}^{4}} \{2 [1 + ρ - 2 ρ^{3}] (- 1 + {(1 - τ)}^{μ + 1})

+ [μ (5 + 12 ρ + 12 ρ^{2}) - 9 - 24 ρ - 12 ρ^{2} - 8 ρ^{3}] τ + [μ^{2} (13 + 12 ρ) - μ (49 + 60 ρ + 24 ρ^{2})] τ^{2}

+ [- μ^{2} (13 + 12 ρ) + 4 μ (11 + 12 ρ + 3 ρ^{2}) + 11 + 26 ρ + 12 ρ^{2} + 4 ρ^{3}] τ^{μ + 1}\}

+ \frac{(2 + 4 ρ) α λ (μ + 3)}{(μ^{2} - 1) {(μ + σ + 1)}^{4}} [1 - τ^{μ + 1} - {(1 - τ)}^{μ + 1}]

+ \frac{(1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{4}} \{2 [1 + ρ - 2 ρ^{3}] (- 1 + {(1 - τ)}^{μ - s + 1})

+ [μ (5 + 12 ρ + 12 ρ^{2}) - 9 - 24 ρ - 12 ρ^{2} - 8 ρ^{3}] τ + [μ^{2} (13 + 12 ρ) - μ (49 + 60 ρ + 24 ρ^{2})] τ^{2}

+ [- μ^{2} (13 + 12 ρ) + 4 μ (11 + 12 ρ + 3 ρ^{2}) + 11 + 26 ρ + 12 ρ^{2} + 4 ρ^{3}] τ^{μ - s + 1}\}

+ \frac{s (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{4}} \{[4 s^{2} + 12 ρ (s - 1) - 3] τ + [μ (12 s - 13) - 4 (s - 1) (5 s + 9 ρ + 8)] τ^{2}

+ [μ^{2} + μ (9 s - 6 ρ - 4) + (s - 1) (8 s + 6 ρ + 11)] τ^{3} + [μ^{2} - μ (3 s - 1) + 2 (s^{2} - 1)] τ^{4}

+ [- 8 μ^{2} + μ (12 s + 5) - 11 + 12 s - 4 s^{2} + 12 sρ - 12 ρ] τ^{μ - s + 1}\}

+ \frac{2 sτ (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{4}} \{[6 μ^{2} - 6 μ (s - 2 ρ - 2) + 7 - 6 s + 2 s^{2} - 6 sρ + 12 ρ + 6 ρ^{2}] τ^{μ - s + 1}

+ [- 8 μ^{2} + μ (12 s + 5) - 11 + 12 s - 4 s^{2} + 12 sρ - 12 ρ] τ^{μ - s + 1}\}

+ \frac{2 sτ (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{4}} \{[6 μ^{2} - 6 μ (s - 2 ρ - 2) + 7 - 6 s + 2 s^{2} - 6 sρ + 12 ρ + 6 ρ^{2}] τ^{μ - s + 1}

- [1 + 2 s^{2} + 6 sρ + 6 ρ^{2}] {(1 - τ)}^{μ - s + 1}\}

+ \frac{(2 + 4 ρ) (1 - α) λ (μ - s + 3)}{({(μ - s)}^{2} - 1) {(μ + σ + 1)}^{4}} [1 - τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}] .

Proof.

We will again only derive the first three moments of $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ for $μ \geq s,$ since the remaining moments given in (3.10) and (3.11) are computed in the same way with the help of Lemma 4.

Now, by employing the definitions given in (3.1), (2.6) and Lemma 3, we obtain equations (3.7), (3.8) and (3.9) as

S_{μ, σ, ρ}^{(α, λ, s)} (1; τ) = (μ + σ + 1) \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) \int_{\frac{κ + ρ}{μ + σ + 1}}^{\frac{κ + ρ + 1}{μ + σ + 1}} dt = (μ + σ + 1) \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) \frac{1}{(μ + σ + 1)}

= \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) = L_{μ, λ}^{(α, s)} (1; τ) = 1,

S_{μ, σ, ρ}^{(α, λ, s)} (t; τ) = (μ + σ + 1) \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) \int_{\frac{κ + ρ}{μ + σ + 1}}^{\frac{κ + ρ + 1}{μ + σ + 1}} tdt = (μ + σ + 1) \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) \frac{2 κ + 2 ρ + 1}{2 (μ + σ + {1)}^{2}}

= \frac{μ}{(μ + σ + 1)} L_{μ, λ}^{(α, s)} (t; τ) + \frac{2 ρ + 1}{2 (μ + σ + 1)} L_{μ, λ}^{(α, s)} (1; τ)

= \frac{2 μτ + 2 ρ + 1}{2 (μ + σ + 1)} + \frac{αλ}{(μ - 1) (μ + σ + 1)} [1 - 2 τ + τ^{μ + 1} - {(1 - τ)}^{μ + 1}]

+ \frac{(1 - α) λ}{(μ - s - 1) (μ + σ + 1)} [1 - 2 τ + τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}],

and

S_{μ, σ, ρ}^{(α, λ, s)} (t^{2}; τ) = (μ + σ + 1) \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) \int_{\frac{κ + ρ}{μ + σ + 1}}^{\frac{κ + ρ + 1}{μ + σ + 1}} t^{2} dt = (μ + σ + 1) \sum_{κ = 0}^{μ} {\tilde{b}}_{μ, κ}^{α, s} (λ; τ) \frac{3 κ^{2} + (3 + 6 ρ) κ + 1 + 3 ρ + 3 ρ^{2}}{3 (μ + σ + {1)}^{3}}

= \frac{μ^{2}}{{(μ + σ + 1)}^{2}} L_{μ, λ}^{(α, s)} (t^{2}; τ) + \frac{(2 ρ + 1) μ}{{(μ + σ + 1)}^{2}} L_{μ, λ}^{(α, s)} (t; τ) + \frac{1 + 3 ρ + 3 ρ^{2}}{3 (μ + σ + {1)}^{2}} L_{μ, λ}^{(α, s)} (1; τ)

= \frac{μ^{2} τ^{2} + (1 + 2 ρ) μτ + [μ + (1 - α) s (s - 1)] τ (1 - τ)}{{(μ + σ + 1)}^{2}} + \frac{1 + 3 ρ + 3 ρ^{2}}{3 (μ + σ + {1)}^{2}}

+ \frac{2 αλ}{(μ - 1) {(μ + σ + 1)}^{2}} [ρ + (μ - 2 ρ - 1) τ - 2 μ τ^{2} + (μ + ρ + 1 + sτ) τ^{μ - s + 1} \break - (ρ + sτ) {(1 - τ)}^{μ - s + 1}],

respectively, which completes the proof.

Corollary 1.

Let $α, τ \in [0, 1],$ $λ \in [- 1, 1],$ and $s \in Z^{+}$ . For $μ < s,$ the central moments of $(α, λ, s)$ -BKSO are as follows:

S_{μ, σ, ρ}^{(α, λ, s)} (t - τ; τ) = \frac{- 2 (σ + 1) τ + 2 ρ + 1}{2 (μ + σ + 1)} + \frac{λ}{(μ - 1) (μ + σ + 1)} [1 - 2 τ + τ^{μ + 1} - {(1 - τ)}^{μ + 1}];

S_{μ, σ, ρ}^{(α, λ, s)} ({(t - τ)}^{2}; τ) = \frac{[- μ + {(σ + 1)}^{2}] τ^{2}}{μ + σ +^{2}} + \frac{[μ - (σ + 1) (1 + 2 ρ)] τ}{μ + σ +^{2}} + \frac{1 + 3 ρ + 3 ρ^{2}}{μ + σ +^{2}} + \frac{2 λ}{(μ - 1) μ + σ +^{2}} [ρ + (μ - 2 ρ - 1) τ - 2 μ τ^{2} \break + (μ + ρ + 1) τ^{μ + 1} - ρ {(1 - τ)}^{μ + 1}] - \frac{2 τλ}{(μ - 1) μ + σ +} [1 - 2 τ + τ^{μ + 1} - {(1 - τ)}^{μ + 1}]

S_{μ, σ, ρ}^{(α, λ, s)} ({(t - τ)}^{3}; τ) = \frac{[μ (5 + 3 σ) - {(σ + 1)}^{3}] τ^{3}}{{(μ + σ + 1)}^{3}} + \frac{3 τ^{2}}{2 (μ + σ + {1)}^{3}} [- μ (5 + 2 σ + 2 ρ) + {(σ + 1)}^{2} (1 + 2 ρ)]

+ \frac{μ (7 + 12 ρ + 6 ρ^{2}) τ}{2 (μ + σ + {1)}^{3}} + \frac{(1 + 4 ρ + 6 ρ^{2} + 4 ρ^{3})}{4 (μ + σ + {1)}^{3}}

+ \frac{- 6 τ (μ + 1) λ}{(μ - 1) {(μ + σ + 1)}^{2}} + \frac{3 τ^{2} λ}{(μ - 1) (μ + σ + 1)} [τ^{μ + 1} - {(1 - τ)}^{μ + 1}]

+ \frac{λ}{2 (μ - 1) {(μ + σ + 1)}^{3}} [- 1 + 7 τ^{μ + 1} + {(1 - τ)}^{μ + 1}]

+ \frac{3 λ}{(μ - 1) {(μ + σ + 1)}^{3}} \{ρ^{2} + [μ - 2 {(ρ + 1)}^{2} - 2 σρ] τ + [- μ + (σ + 1) (σ + 4 ρ + 3)] τ^{2}

+ 2 [μ - {(σ + 1)}^{2}] τ^{3} + {(μ + ρ + 1)}^{2} τ^{μ + 1} - ρ^{2} {(1 - τ)}^{μ + 1}\}

+ \frac{(μ + 3) λ}{(μ^{2} - 1) {(μ + σ + 1)}^{3}} [1 - τ^{μ + 1} - {(1 - τ)}^{μ + 1}];

S_{μ, σ, ρ}^{(α, λ, s)} ({(t - τ)}^{4}; τ) = \frac{τ^{4}}{{(μ + σ + 1)}^{4}} \{3 μ^{2} - 2 [3 σ^{2} + 10 σ + 10] μ + {(σ + 1)}^{4}\}

+ \frac{2 τ^{3}}{{(μ + σ + 1)}^{4}} \{- 3 μ^{2} + [3 σ^{2} + 3 σ (5 + 2 ρ) + 10 (2 + ρ)] μ - {(σ + 1)}^{3} (1 + 2 ρ)\}

+ \frac{τ^{2}}{{(μ + σ + 1)}^{4}} \{3 μ^{2} - [6 ρ^{2} + (5 + 2 σ) (5 + 6 ρ)] μ + 2 {(σ + 1)}^{2} (1 + 3 ρ + 3 ρ^{2})\}

+ \frac{τ}{{(μ + σ + 1)}^{4}} \{[6 ρ^{2} + 10 ρ + 5] μ - (σ + 1) (1 + 4 ρ + 6 ρ^{2} + 4 ρ^{3})\}

+ \frac{1 + 5 ρ + 10 ρ^{2} + 10 ρ^{3} + 5 ρ^{4}}{5 (μ + σ + {1)}^{4}} - \frac{24 μτλ τ^{μ + 1}}{{(μ + σ + 1)}^{3}}

+ \frac{λ}{{(μ + σ + 1)}^{4}} \{τ + [6 μ^{2} - μ] τ^{2} - 2 [μ^{2} + 2 μ (11 + 6 ρ)] τ^{3}

+ 8 [2 μ^{2} + μ (5 + 3 σ)] τ^{4} + [4 μ^{2} + μ (29 + 24 ρ) - 1] τ^{μ + 1}\}

+ \frac{4 τ^{3} λ}{(μ - 1) (μ + σ + 1)} [- τ^{μ + 1} + {(1 - τ)}^{μ + 1}] + \frac{12 τ^{2} λ}{(μ - 1) {(μ + σ + 1)}^{2}} [(μ + ρ + 1) τ^{μ + 1} - ρ {(1 - τ)}^{μ + 1}]

+ \frac{2 τλ}{(μ - 1) {(μ + σ + 1)}^{3}} \{- (1 - 6 ρ^{2}) {(1 - τ)}^{μ + 1} + [6 {(μ - 1)}^{2} - 6 {(ρ + 1)}^{2} \break - 12 μ (ρ + 1) - 13] τ^{μ + 1}\}

+ \frac{(2 + 2 ρ - 4 ρ^{3}) λ}{(μ - 1) {(μ + σ + 1)}^{4}} [- 1 - τ^{μ + 1} + {(1 - τ)}^{μ + 1}]

+ \frac{λ}{(μ - 1) {(μ + σ + 1)}^{4}} \{8 [- 2 μ^{3} - 3 μ^{2} (σ + 1) + {(σ + 1)}^{3}] τ^{4}

- 4 [μ^{3} - 6 μ^{2} (ρ + 2) - 9 μ (σ + 1) + σ {(σ + 3)}^{2} + 6 ρ {(σ + 1)}^{2} + 4] τ^{3}

+ [μ^{2} - μ (37 + 12 σ + 36 ρ) + 24 σ {(ρ + 1)}^{2} + 24 (ρ^{2} + 1) + 12 ρ (σ^{2} + 3)] τ^{2}

+ [μ (7 + 12 ρ) + 2 σ (1 - 6 ρ^{2}) - 8 ρ (ρ^{2} + 3 ρ + 3) - 7] τ

+ [- μ^{2} (13 + 12 ρ) + 4 μ (11 + 12 ρ + 3 ρ^{2}) + 13 + 28 ρ + 12 ρ^{2}] τ^{μ + 1}\}

+ \frac{4 τ (μ + 3) λ}{(μ^{2} - 1) {(μ + σ + 1)}^{3}} [- 1 + τ^{μ + 1} + {(1 - τ)}^{μ + 1}] + \frac{(2 + 4 ρ) (μ + 3) λ}{(μ^{2} - 1) {(μ + σ + 1)}^{4}} [1 - τ^{μ + 1} - {(1 - τ)}^{μ + 1}] .

Corollary 2.

Let $α, τ \in [0, 1],$ $λ \in [- 1, 1],$ and $s \in Z^{+}$ . For $μ \geq s,$ the central moments of $(α, λ, s)$ -BKS operators are as follows

S_{μ, σ, ρ}^{(α, λ, s)} (t - τ; τ) = \frac{- 2 (σ + 1) τ + 2 ρ + 1}{2 (μ + σ + 1)} + \frac{αλ}{(μ - 1) (μ + σ + 1)} [1 - 2 τ + τ^{μ + 1} - {(1 - τ)}^{μ + 1}]

(31)

+ \frac{(1 - α) λ}{(μ - s - 1) (μ + σ + 1)} [1 - 2 τ + τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}];

(31)

S_{μ, σ, ρ}^{(α, λ, s)} ({(t - τ)}^{2}; τ) = \frac{[- μ + {(σ + 1)}^{2} - (1 - α) s (s - 1)] τ^{2}}{{(μ + σ + 1)}^{2}} + \frac{[μ - (σ + 1) (1 + 2 ρ) + (1 - α) s (s - 1)] τ}{{(μ + σ + 1)}^{2}} + \frac{1 + 3 ρ + 3 ρ^{2}}{3 (μ + σ + {1)}^{2}}

+ \frac{2 αλ}{(μ - 1) {(μ + σ + 1)}^{2}} [ρ + (μ - 2 ρ - 1) τ - 2 μ τ^{2} + (μ + ρ + 1) τ^{μ + 1} - ρ {(1 - τ)}^{μ + 1}]

+ \frac{2 (1 - α) λ}{(μ - s - 1) (μ + σ + 1)} [ρ + (μ - 2 ρ - 1) τ - 2 μ τ^{2} + (μ + ρ + 1) τ^{μ - s + 1} - ρ {(1 - τ)}^{μ - s + 1}]

+ \frac{2 sτ (1 - α) λ}{(μ - s - 1) (μ + σ + 1)} [τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}] - \frac{2 ταλ}{(μ - 1) (μ + σ + 1)} [1 - 2 τ + τ^{μ + 1} - {(1 - τ)}^{μ + 1}]

- \frac{2 τ (1 - α) λ}{(μ - s - 1) (μ + σ + 1)} [1 - 2 τ + τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}];

S_{μ, σ, ρ}^{(α, λ, s)} ({(t - τ)}^{3}; τ) = \frac{[μ (5 + 3 σ) - {(σ + 1)}^{3} + (1 - α) s (s - 1) (5 + 3 σ + 2 s)] τ^{3}}{{(μ + σ + 1)}^{3}}

+ \frac{3 τ^{2}}{2 (μ + σ + {1)}^{3}} [- μ (5 + 2 σ + 2 ρ) + {(σ + 1)}^{2} (1 + 2 ρ) - (1 - α) s (s - 1) (5 + 2 σ + 2 ρ + 2 s)]

+ \frac{[μ (7 + 12 ρ + 6 ρ^{2}) + (1 - α) s (s - 1) (5 + 6 ρ + 2 s)] τ}{2 (μ + σ + {1)}^{3}} + \frac{(1 + 4 ρ + 6 ρ^{2} + 4 ρ^{3})}{4 (μ + σ + {1)}^{3}}

+ \frac{- 6 τ (μ + 1) α λ}{(μ - 1) {(μ + σ + 1)}^{2}} + \frac{3 τ^{2} αλ}{(μ - 1) (μ + σ + 1)} [τ^{μ + 1} - {(1 - τ)}^{μ + 1}]

+ \frac{αλ}{2 (μ - 1) {(μ + σ + 1)}^{3}} [- 1 + 7 τ^{μ + 1} + {(1 - τ)}^{μ + 1}]

+ \frac{3 αλ}{(μ - 1) {(μ + σ + 1)}^{3}} \{ρ^{2} + [μ - 2 {(ρ + 1)}^{2} - 2 σρ] τ + [- μ + (σ + 1) (σ + 4 ρ + 3)] τ^{2}

+ 2 [μ - {(σ + 1)}^{2}] τ^{3} + {(μ + ρ + 1)}^{2} τ^{μ + 1} - ρ^{2} {(1 - τ)}^{μ + 1}\}

+ \frac{(μ + 3) α λ}{(μ^{2} - 1) {(μ + σ + 1)}^{3}} [1 - τ^{μ + 1} - {(1 - τ)}^{μ + 1}] + \frac{3 τ^{2} (1 - α) λ}{(μ - s - 1) (μ + σ + 1)} [τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}]

+ \frac{6 s τ^{2} (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{2}} [- τ^{μ - s + 1} + {(1 - τ)}^{μ - s + 1}]

+ \frac{6 τ (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{2}} [- (μ + ρ + 1) τ^{μ - s + 1} + ρ {(1 - τ)}^{μ - s + 1}]

+ \frac{3 (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{3}} \{[μ + s (s - 1) - 2 {(ρ + 1)}^{2} - 2 σρ] τ

+ [- 3 μ - 3 s (s - 1) + (σ + 1) (σ + 4 ρ + 3)] τ^{2} + 2 [μ + s (s - 1) - {(σ + 1)}^{2}] τ^{3}

+ [{(μ - s + ρ + 1)}^{2} + s (2 ρ + 1)] τ^{μ - s + 1}\}

+ \frac{3 sτ (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{3}} \{[2 (μ + ρ + 1) - s] τ^{μ - s + 1} - [s + 2 ρ] {(1 - τ)}^{μ - s + 1}\}

+ \frac{(1 - α) λ}{2 (μ - s - 1) {(μ + σ + 1)}^{3}} [- 1 + 6 ρ^{2} + 7 τ^{μ - s + 1} + (1 - 6 ρ^{2}) {(1 - τ)}^{μ - s + 1}]

+ \frac{(μ - s + 3) (1 - α) λ}{({(μ - s)}^{2} - 1) {(μ + σ + 1)}^{3}} [1 - τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}];

(33)

S_{μ, σ, ρ}^{(α, λ, s)} ({(t - τ)}^{4}; τ) = \frac{τ^{4}}{{(μ + σ + 1)}^{4}} {3 μ^{2} - 2 [3 σ^{2} + 10 σ + 10] μ + {(σ + 1)}^{4}

(33)

+ (1 - α) s (s - 1) [6 μ - 3 {(s + 1)}^{2} - 6 {(σ + 1)}^{2} - (s + 1) (8 σ + 11)]}

+ \frac{2 τ^{3}}{{(μ + σ + 1)}^{4}} \{- 3 μ^{2} + [3 σ^{2} + 3 σ (5 + 2 ρ) + 10 (2 + ρ)] μ - {(σ + 1)}^{3} (1 + 2 ρ)

+ (1 - α) s (s - 1) [- 6 μ + 3 (s^{2} + σ^{2}) + s (17 + 6 σ + 4 ρ) + 3 σ (5 + 2 ρ) + 10 (2 + ρ)]\}

+ \frac{τ^{2}}{{(μ + σ + 1)}^{4}} \{3 μ^{2} - [6 ρ^{2} + (5 + 2 σ) (5 + 6 ρ)] μ + 2 {(σ + 1)}^{2} (1 + 3 ρ + 3 ρ^{2})

+ (1 - α) s (s - 1) [6 μ - 10 s - 6 (3 + 4 ρ) - (s + σ) (10 + 12 ρ + 4 s)]\}

+ \frac{τ}{{(μ + σ + 1)}^{4}} \{[6 ρ^{2} + 10 ρ + 5] μ - (σ + 1) (1 + 4 ρ + 6 ρ^{2} + 4 ρ^{3})

+ (1 - α) s (s - 1) [s^{2} + s (3 + 4 ρ) + 6 ρ^{2} + 10 ρ + 5]\}

+ \frac{1 + 5 ρ + 10 ρ^{2} + 10 ρ^{3} + 5 ρ^{4}}{5 (μ + σ + {1)}^{4}}

+ \frac{λ}{{(μ + σ + 1)}^{4}} \{τ + [6 μ^{2} - μ] τ^{2} - 2 [μ^{2} + 2 μ (11 + 6 ρ)] τ^{3} + 8 [2 μ^{2} + μ (5 + 3 σ)] τ^{4}

+ α [4 μ^{2} + μ (29 + 24 ρ) - 1] τ^{μ + 1} + (1 - α) [4 μ^{2} + μ (29 + 24 ρ) - 1] τ^{μ - s + 1}\}

+ \frac{- 24 μτλ}{{(μ + σ + 1)}^{3}} [α τ^{μ + 1} + (1 - α) τ^{μ - s + 1}]

+ \frac{4 τ^{3} αλ}{(μ - 1) (μ + σ + 1)} [- τ^{μ + 1} + {(1 - τ)}^{μ + 1}] + \frac{12 τ^{2} αλ}{(μ - 1) {(μ + σ + 1)}^{2}} [(μ + ρ + 1) τ^{μ + 1} - ρ {(1 - τ)}^{μ + 1}]

+ \frac{2 ταλ}{(μ - 1) {(μ + σ + 1)}^{3}} \{- (1 - 6 ρ^{2}) {(1 - τ)}^{μ + 1} + [6 {(μ - 1)}^{2} - 6 {(ρ + 1)}^{2} \break - 12 μ (ρ + 1) - 13] τ^{μ + 1}\}

+ \frac{(2 + 2 ρ - 4 ρ^{3}) αλ}{(μ - 1) {(μ + σ + 1)}^{4}} [- 1 - τ^{μ + 1} + {(1 - τ)}^{μ + 1}]

+ \frac{αλ}{(μ - 1) {(μ + σ + 1)}^{4}} \{8 [- 2 μ^{3} - 3 μ^{2} (σ + 1) + {(σ + 1)}^{3}] τ^{4}

- 4 [μ^{3} - 6 μ^{2} (ρ + 2) - 9 μ (σ + 1) + σ {(σ + 3)}^{2} + 6 ρ {(σ + 1)}^{2} + 4] τ^{3}

+ [μ^{2} - μ (37 + 12 σ + 36 ρ) + 24 σ {(ρ + 1)}^{2} + 24 (ρ^{2} + 1) + 12 ρ (σ^{2} + 3)] τ^{2}

+ [μ (7 + 12 ρ) + 2 σ (1 - 6 ρ^{2}) - 8 ρ (ρ^{2} + 3 ρ + 3) - 7] τ

+ [- μ^{2} (13 + 12 ρ) + 4 μ (11 + 12 ρ + 3 ρ^{2}) + 13 + 28 ρ + 12 ρ^{2}] τ^{μ + 1}\}

+ \frac{4 τ (μ + 3) α λ}{(μ^{2} - 1) {(μ + σ + 1)}^{3}} [- 1 + τ^{μ + 1} + {(1 - τ)}^{μ + 1}] + \frac{(2 + 4 ρ) (μ + 3) α λ}{(μ^{2} - 1) {(μ + σ + 1)}^{4}} [1 - τ^{μ + 1} - {(1 - τ)}^{μ + 1}]

+ \frac{4 τ^{3} (1 - α) λ}{(μ - s - 1) (μ + σ + 1)} [- τ^{μ - s + 1} + {(1 - τ)}^{μ - s + 1}]

+ \frac{12 τ^{2} (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{2}} [(μ + ρ + 1) τ^{μ - s + 1} - ρ {(1 - τ)}^{μ - s + 1}]

+ \frac{2 τ (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{3}} \{- (1 - 6 ρ^{2}) {(1 - τ)}^{μ - s + 1}

+ [6 {(μ - 1)}^{2} - 6 {(ρ + 1)}^{2} - 12 μ (ρ + 1) - 13] τ^{μ - s + 1}\}

+ \frac{(2 + 2 ρ - 4 ρ^{3}) (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{4}} [- 1 - τ^{μ - s + 1} + {(1 - τ)}^{μ - s + 1}]

+ \frac{(1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{4}} \{8 [- 2 μ^{3} - 3 μ^{2} (σ + 1) + {(σ + 1)}^{3}] τ^{4}

- 4 [μ^{3} - 6 μ^{2} (ρ + 2) - 9 μ (σ + 1) + σ {(σ + 3)}^{2} + 6 ρ {(σ + 1)}^{2} + 4] τ^{3}

+ [μ^{2} - μ (37 + 12 σ + 36 ρ) + 24 σ {(ρ + 1)}^{2} + 24 (ρ^{2} + 1) + 12 ρ (σ^{2} + 3)] τ^{2}

+ [μ (7 + 12 ρ) + 2 σ (1 - 6 ρ^{2}) - 8 ρ (ρ^{2} + 3 ρ + 3) - 7] τ

+ [- μ^{2} (13 + 12 ρ) + 4 μ (11 + 12 ρ + 3 ρ^{2}) + 13 + 28 ρ + 12 ρ^{2}] τ^{μ - s + 1}\}

+ \frac{12 s τ^{3} (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{2}} [τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}]

+ \frac{12 s τ^{2} (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{3}} [(- 2 μ + s - 2 ρ - 2) τ^{μ - s + 1} + (s + 2 ρ) {(1 - τ)}^{μ - s + 1}]

+ \frac{2 sτ (1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{4}} \{- [2 s^{2} + 6 ρ (s + ρ) + 1] {(1 - τ)}^{μ - s + 1}

+ [6 μ^{2} + 6 μ (- 2 s + 2 ρ + 3) + 2 s^{2} - 6 s (σ + ρ + 2) + 6 ρ (ρ + 2) + 6 σ + 13] τ^{μ - s + 1}\}

+ \frac{(1 - α) λ}{(μ - s - 1) {(μ + σ + 1)}^{4}} \{[4 s^{2} + 12 ρ (s - 1) - 3] τ + [- μ - 4 s (5 s + 6) \break - 12 (s - 1) (σ + 3 ρ) + 30] τ^{2}

+ 4 [μ^{2} - μ (6 ρ + 11) + 8 s^{2} + 3 s (3 σ + 2 ρ + 4) - (9 σ + 6 ρ + 20)] τ^{3}

+ 8 [2 μ^{2} + μ (3 σ + 5) - (s - 1) (2 s + 3 σ + 5)] τ^{4}

+ [- 8 μ^{2} + μ (12 s + 5) - 4 s (s - 3) + 12 ρ (s - 1) - 11] τ^{μ - s + 1}\}

+ \frac{4 τ (μ - s + 3) (1 - α) λ}{({(μ - s)}^{2} - 1) {(μ + σ + 1)}^{3}} [- 1 - τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}]

+ \frac{(2 + 4 ρ) (μ - s + 3) (1 - α) λ}{({(μ - s)}^{2} - 1) {(μ + σ + 1)}^{4}} [1 - τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}] .

Lemma 7.

Let $∥\cdot∥$ be the uniform norm defined on $[0, 1]$ and $α, τ \in [0, 1],$ $λ \in [- 1, 1] .$ Then for a given $ξ \in C [0, 1]$

∥S_{μ, σ, ρ}^{(α, λ, s)} ξ∥ \leq ∥ξ∥,

for all $μ \in N$ and $s \in Z^{+} .$

Proof.

By taking into account Lemma 5, Lemma 6 and the definition of $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ given in (3.1), we have

∥S_{μ, σ, ρ}^{(α, λ, s)} ξ∥ \leq ∥ξ∥ S_{μ, σ, ρ}^{(α, λ, s)} (1; τ) = ∥ξ∥,

which concludes the proof.□

Theorem 2.

If $ξ$ is continuous on $[0, 1]$ , then

lim_{μ \to \infty} S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ) = ξ (τ)

uniformly on $[0, 1] .$

Proof.

Utilizing the well-known Bohman-Korovkin theorem (Korovkin Citation1960, Bohman Citation1952), our objective is to prove the subsequent condition for uniform convergence:

lim_{μ \to \infty} S_{μ, σ, ρ}^{(α, λ, s)} (e_{i}; τ) = τ^{i},

where

e_{i} = τ^{i}

and

i = 0, 1, 2.

By (3.7), (3.8) and (3.9) the following results are satisfied:

lim_{μ \to \infty} S_{μ, σ, ρ}^{(α, λ, s)} (e_{0}; τ) = 1, lim_{μ \to \infty} S_{μ, σ, ρ}^{(α, λ, s)} (e_{1}; τ) = τ lim_{μ \to \infty} S_{μ, σ, ρ}^{(α, λ, s)} (e_{2}; τ) = τ^{2}

and the proof is completed.

Before we proceed with the following theorem, we shall note that $C^{2} [0, 1]$ refers to the set of functions $ξ \in C [0, 1],$ which are twice differentiable with $ξ^{'}, ξ^{^{''}} \in C [0, 1]$ .

Theorem 3.

Assume that $ξ \in C [0, 1] .$ If $ξ^{^{''}} (τ)$ exists for a point $τ \in [0, 1],$ then

lim_{μ \to \infty} \{μ S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ) - ξ (τ)]\} = \frac{- 2 (σ + 1) τ + 2 ρ + 1}{2} ξ^{^{'}} (τ) + \frac{τ - τ^{2}}{2} ξ^{^{^{''}}} (τ),

uniformly on $[0, 1] .$

Proof.

Consider the case $μ \geq s .$ For a fixed $τ \in [0, 1]$ suppose that $ξ \in C^{2} [0, 1]$ . Due to Taylor’s expansion with a reminder in Peano’s form, we can write

(34)

ξ (t) - ξ (τ) = (t - τ) ξ^{'} (τ) + \frac{1}{2} (t - τ)^{2} ξ^{^{''}} (τ) + (t - τ)^{2} {\tilde{r}}_{τ} (t),

(34)

where

{\tilde{r}}_{τ} (t)

is the remainder term satisfying

{\tilde{r}}_{τ} (t) \in C [0, 1]

and

lim_{t \to τ} {\tilde{r}}_{τ} (t) = 0

. By adopting identity (3.18) for

(α, λ, s)

-BKSO, we obtain

(35)

S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ) - ξ (τ) = ξ^{'} (τ) S_{μ, σ, ρ}^{(α, λ, s)} (t - τ; τ) + \frac{ξ^{^{''}} (τ)}{2} S_{μ, σ, ρ}^{(α, λ, s)} ({(t - τ)}^{2}; τ) + S_{μ, σ, ρ}^{(α, λ, s)} ({(t - τ)}^{2} {\tilde{r}}_{τ} (t); τ) .

(35)

Multiplication of both sides of (3.19) by $μ$ and utilization of the Cauchy-Schwarz inequality, respectively, yield

μ S_{μ, σ, ρ}^{(α, λ, s)} ((t - τ)^{2} ξ_{τ} (t); τ) \leq \sqrt{μ^{2} S_{μ, σ, ρ}^{(α, λ, s)} ({(t - τ)}^{4}; τ)} \sqrt{S_{μ, σ, ρ}^{(α, λ, s)} ({\tilde{r}}_{τ} (t); τ)} .

Consequently, by fourth order central moment (34) and Theorem 2, and the following expression

lim_{μ \to \infty} μ^{2} S_{μ, σ, ρ}^{(α, λ, s)} ({(t - τ)}^{4}; τ) = 3 τ^{4} - 2 λ (1 + 2 α) τ^{3} - 6 τ^{3} + 3 (2 λ + 1) τ^{2},

we have

lim_{μ \to \infty} μ [S_{μ, σ, ρ}^{(α, λ, s)} ((t - τ)^{2} {\tilde{r}}_{τ} (t); τ)] = 0.

Hence, by (3.12), (3.13), (3.15) and (3.16), the proof is completed for $μ \geq s .$ Using the same techniques and following likewise steps, one can prove the statement for $μ < s$ ; therefore, it is omitted.□

4. Convergence Results for Kantorovich-Stancu Type (α, λ, s)-Bernstein Operators $S_{μ, σ, ρ}^{(α, λ, s)}$

In this section, we investigate the convergence characteristics and a Voronovskaja type approximation result for $(α, λ, s)$ -BKSO by utilizing the weighted $B$ -statistical convergence notion. In addition, we give an estimation for the rate of the weighted $B$ -statistical convergence of these operators. For the readers who are not familiar with the concepts of asymptotic density, statistical convergence, weighted statistical convergence, $B$ -summability, regular summability matrix, $B$ -statistically convergence, and weighted $B$ -statistically convergence, we refer to pioneering works (Kadak et al. Citation2017; Kadak Citation2020) and the references therein. The application of Bohmann-Korovkin theorem using the weighted statistical convergence is studied in the papers (Mursaleen et al. Citation2012; Belen and Mohiuddine Citation2013; Mohiuddine Citation2016; Baliarsingh et al. Citation2018; Kadak Citation2018).

We begin our work by considering the set $M$ , a subset of $N_{0} := N \cup {0}$ . The asymptotic density of the set $M$ is denoted by $D (M)$ and is defined by

D (M) = lim_{μ \to \infty} \frac{1}{μ} {m = xμ : m \in M}

provided the limit exists. Note that asymptotic density lets us define the statistical convergence. Consequently, we say a sequence $τ = \{τ_{m}\}$ is statistically convergent to the number $L$ if

lim_{μ \to \infty} \frac{1}{μ} {m : m = xμ and | τ_{m} - L | = xε} = 0, for all ε 0.

Next, consider a sequence of non-negative numbers denoted $η = \{η_{m}\}$ given with

(36)

η_{0} > 0 and T_{μ} = \sum_{m = 0}^{μ} η_{m} \to \infty as μ \to \infty .

(36)

lim_{μ \to \infty} \frac{1}{n_{μ}} {m = x n_{μ} : η_{m} | τ_{m} - L | = xε} = 0, for all ε 0,

then we say $τ = \{τ_{m}\}$ is weighted statistically convergent to the number $L$ . Kolk (Kolk Citation1993) introduced a new matrix method, so-called $B$ -summability as follows:

Let $B = \{B_{j}\} = \{{\tilde{b}}_{μm} (j)\}$ be a sequence of infinite matrices. If

lim_{μ \to \infty} (B_{j} τ)_{μ} = lim_{μ \to \infty} \sum_{m} {\tilde{b}}_{μm} (j) η_{m} = B - lim τ uniformly

for $j = 0, 1, 2, \dots$ , then we say $τ \in ℓ_{\infty}$ is $B$ -summable to the value $B - lim τ$ . The method $B = \{B_{j}\}$ is said to be regular if and only if the below conditions are satisfied (see (Stieglitz Citation1973; Bell Citation1973)):C1: $∥ B ∥= sup_{μ, j} \sum_{m} | {\tilde{b}}_{μm} (j) | < \infty;$

C2: $lim_{μ \to \infty} {\tilde{b}}_{μm} (j) = 0 uniformly in j$ for $\forall_{m} \in N_{0}$ ;

C3: $lim_{μ \to \infty} \sum_{m} {\tilde{b}}_{μm} (j) = 1 uniformly in j .$

The set of regular methods $B$ satisfying ${\tilde{b}}_{μm} (j) = x 0$ for all $μ$ , $m$ , and $j,$ is denoted by $ℜ^{+}$ . For a given $B \in ℜ^{+}$ , if

lim_{μ \to \infty} \sum m : | τ_{m} - L | = xε {\tilde{b}}_{μm} (j) = 0 uniformly in j

for all $ε > 0$ , then we say $τ = \{τ_{m}\}$ is $B$ -statistically convergent to the number $L$ .

Definition 1

([33]). Assume that $B = {\{B_{j}\}}_{j \in N} \in ℜ^{+}$ and let sequence $η = \{η_{m}\}$ be defined as in (37). If

lim_{r \to \infty} \frac{1}{n_{r}} \sum_{μ = 0}^{r} η_{μ} \sum m : | τ_{m} - L | = xε {\tilde{b}}_{μm} (j) = 0 uniformly in j

for all $m$ and $ε > 0$ , then the sequence $τ = \{τ_{μ}\}$ is weighted $B$ -statistically convergent to the number $L$ . In symbols, it is written as $[sta t_{B}, η_{μ}] - lim τ = L$ .

Theorem 4.

Let $α, τ \in [0, 1],$ $λ \in [- 1, 1],$ $ξ \in C [0, 1]$ and $B \in ℜ^{+}$ . Then we have

[sta t_{B}, η_{μ}] - lim_{μ \to \infty} {∥S_{μ, σ, ρ}^{(α, λ, s)} ξ - ξ∥}_{C [0, 1]} = 0,

for all $μ$ and positive integer $s .$

Proof.

For $μ \geq s$ , let $ξ \in C [0, 1]$ and consider a sequence of functions $e_{i} (τ) = τ^{i}$ for a fixed $τ \in [0, 1] .$ It is enough to show that

[sta t_{B}, η_{μ}] - lim_{μ \to \infty} {∥S_{μ, σ, ρ}^{(α, λ, s)} e_{i} - e_{i}∥}_{C [0, 1]} = 0, for i = 0, 1, 2,

on the grounds of Korovkin-Bohman Theorem (see (Bohman Citation1952; Korovkin Citation1960)). Accordingly for $i = 0,$ we obtain

(37)

[sta t_{B}, η_{μ}] - lim_{μ \to \infty} {∥S_{μ, σ, ρ}^{(α, λ, s)} e_{0} - e_{0}∥}_{C [0, 1]} = 0,

(37)

by employing identity (3.7) from Lemma 6. Now for $i = 1,$ by substitution of identities (3.8) and (3.15) respectively, we get

sup_{τ \in [0, 1]} |S_{μ, σ, ρ}^{(α, λ, s)} (e_{1}; τ) - e_{1} (τ)| = sup_{τ \in [0, 1]} | \frac{2 μτ + 2 ρ + 1}{μ + σ +} + \frac{αλ}{(μ - 1) μ + σ +} [1 - 2 τ + τ^{μ + 1} - {(1 - τ)}^{μ + 1}] + \frac{(1 - α) λ}{(μ - s - 1) μ + σ +} [1 - 2 τ + τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}] - τ | sup_{τ \in [0, 1]} |S_{μ, σ, ρ}^{(α, λ, s)} (t - ττ)|

which implies

sup_{τ \in [0, 1]} |S_{μ, σ, ρ}^{(α, λ, s)} (e_{1}; τ) - e_{1} (τ)| \leq \frac{1 + 2 ρ}{2 μ + 2 σ + 2} .

Next, for a fixed $\frac{n}{ε} 0,$ we choose a number $ε > 0$ such that $ε \frac{n}{ε}$ . Subsequently, defining the sets $G$ and $G_{1}$ as follows

n := μ \in n : ∥n_{μ, σ, ρ}^{(α, λ, s)} n_{1} - n_{1}∥ = x \frac{n}{ε}

n_{1} := μ \in n : \frac{1 + 2 ρ}{2 μ + 2 σ + 2} . = x \frac{n}{ε} - ε,

we attain

(38)

\frac{1}{T_{r}} \sum_{μ = 0}^{r} η_{μ} \sum_{m \in G} {\tilde{b}}_{μm} (j) \leq \frac{1}{T_{r}} \sum_{μ = 0}^{r} η_{μ} \sum_{m \in G_{1}} {\tilde{b}}_{μm} (j) .

(38)

Taking $r \to \infty$ in inequality (4.3) yields

(39)

[sta t_{B}, η_{μ}] - lim_{μ \to \infty} {∥S_{μ, σ, ρ}^{(α, λ, s)} e_{1} - e_{1}∥}_{C [0, 1]} = 0.

(39)

Last, for $i = 2,$ we can write

sup_{τ \in [0, 1]} |S_{μ, σ, ρ}^{(α, λ, s)} (e_{1} τ) - e_{1} (τ)| = sup_{τ \in [0, 1]} | \frac{2 μτ + 2 ρ + 1}{μ + σ +} + \frac{αλ}{(μ - 1) μ + σ +} [1 - 2 τ + τ^{μ + 1} - {(1 - τ)}^{μ + 1}] + \frac{(1 - α) λ}{(μ - s - 1) μ + σ +} [1 - 2 τ + τ^{μ - s + 1} - {(1 - τ)}^{μ - s + 1}] - τ | sup_{τ \in [0, 1]} |S_{μ, σ, ρ}^{(α, λ, s)} (t - ττ)|

by utilizing identities (3.9) and (3.16) respectively. In a similar fashion to the preceding case, we can deduce that

(40)

[sta t_{B}, η_{μ}] - lim_{μ \to \infty} {∥S_{μ, σ, ρ}^{(α, λ, s)} e_{2} - e_{2}∥}_{C [0, 1]} = 0.

(40)

Hence, the proof is complete for $μ \geq s$ due equations (4.2), (4.4) and (4.5). The proof of the assertion for $μ < s$ is omitted since it is tantamount to the proof of the case $μ \geq s$ .□

Definition 2

([24]). Assume that $B \in ℜ^{+}$ . If

lim_{i} \frac{1}{i} r = xi : |\frac{1}{n_{r}} \sum_{μ = 0}^{r} η_{μ} \sum_{m = 1}^{\infty} τ_{μ} {\tilde{b}}_{μm} (j) - n| = xε = 0 uniformly inj,

for all $ε > 0$ , then we say $τ = \{τ_{μ}\}$ is statistically weighted $B$ -summable to the limit value $L$ . In symbols, it is written as ${\bar{n}}_{B} (stat) - limτ = n$ .

Theorem 5

([24]). A bounded sequence $τ = \{τ_{μ}\}$ , $∀μ \in N$ , is statistically weighted $B$ -summable to the number $L$ if it is weighted $B$ -statistically convergent to the same number $L$ . However, the reverse is not true.

We deduce the next corollary due to Theorem 4 and Theorem 5.

Corollary 3.

For $ξ \in C [0, 1]$ and $B \in ℜ^{+}$ , we have

{\bar{n}}_{B} (stat) - \lim {‖ n_{μ, σ, ρ}^{(α, λ, s)} ξ - ξ ‖}_{C [0, 1]} = 0.

Now, we are ready to give an estimation for the rate of the weighted $B$ -statistical convergence of $(α, λ, s)$ -BKSO to $ξ \in C [0, 1]$ via modulus of continuity of first order. First, we shall present the definition regarding the weighted $B$ -statistical convergence of positive and non-decreasing sequences, which we will utilize in the theorem that follows.

Definition 3

([24]). Let $B \in ℜ^{+}$ and assume that $\{ψ_{m}\}$ is a non-decreasing and positive sequence. If

lim_{r \to \infty} \frac{1}{ψ_{r} n_{r}} \sum_{μ = 0}^{r} η_{μ} \sum m : | τ_{m} - n | = xε {\tilde{b}}_{μm} (j) = 0 uniformly inj,

for any $ε > 0$ , then we say sequence $τ = \{τ_{m}\}$ is weighted $B$ -statistically convergent to limit $L$ with the rate $o (ψ_{m})$ . In symbols, it is written as $[sta t_{B}, η_{μ}] - o (ψ_{m}) = τ_{m} - L .$

Theorem 6.

Let $B = {\tilde{b}}_{μm} (j)$ be a non-negative regular summability matrix and ${\{φ_{μ}\}}_{μ \in N}$ be a non-decreasing and positive sequence. If the condition

$ω (ξ; ρ_{μ}) = [sta t_{B}, η_{μ}] - o (φ_{μ})$ on $[0, 1]$ , where $ρ_{μ} := {∥S_{μ, σ, ρ}^{(α, λ, s)} {(t - τ)}^{2}∥}_{C [0, 1]}^{1 / 2}$ with $t \in 0, 1],$ holds true, then for $ξ \in C [0, 1]$

{∥S_{μ, σ, ρ}^{(α, λ, s)} ξ - ξ∥}_{C [0, 1]} = [sta t_{B}, η_{μ}] - o (φ_{μ}),

where $ω$ is the usual modulus of continuity.

Proof.

For fixed $τ \in [0, 1]$ assume that $ξ (τ) \in C [0, 1]$ . Linearity of $(α, λ, s)$ -BKSO implies

(41)

|S_{μ, σ, ρ}^{(α, λ, s)} (ξ (t); τ) - ξ (τ)| \leq_{μ, σ, ρ}^{(α, λ, s)} (|ξ (t) - ξ (τ)|; τ) + N | S_{μ, σ, ρ}^{(α, λ, s)} (e_{0}; τ) - e_{0} (τ) |

(41)

\leq ω (ξ, y) S_{μ, σ, ρ}^{(α, λ, s)} (\frac{| t - τ |}{y} + 1; τ)

= ω (ξ, y) \{S_{μ, σ, ρ}^{(α, λ, s)} (e_{0}; τ) + \frac{1}{y^{2}} S_{μ, σ, ρ}^{(α, λ, s)} ((t - τ)^{2}; τ)\},

where $N = {sup}_{τ \in [0, 1]} |ξ (τ)| .$ Subsequently, considering the supremum over $[0, 1]$ for (4.6) and letting $y = ρ_{μ}$ , one obtains

(42)

{∥S_{μ, σ, ρ}^{(α, λ, s)} ξ (t) - ξ (τ)∥}_{C [0, 1]} \leq ω (ξ, ρ_{μ}) \{\frac{1}{ρ_{μ}^{2}} {∥S_{μ, σ, ρ}^{(α, λ, s)} {(t - τ)}^{2}∥}_{C [0, 1]} + 1\} = 2 ω (ξ, ρ_{μ}) .

(42)

The following sets are defined or a given $ε > 0$ as

H = \{μ : {∥S_{μ, σ, ρ}^{(α, λ, s)} ξ (t) - ξ (τ)∥}_{C [0, 1]} \geq ε\},

H_{1} = \{μ : ω (ξ, ρ_{μ}) \geq \frac{ε}{2}\} .

Here one may easily verify that

\frac{1}{φ_{μ}} \sum_{ζ \in H} b_{μζ} \leq \frac{1}{φ_{μ}} \sum_{ζ \in H_{1}} b_{μζ} .

Last, by employing the given condition the theorem is proved.□

Our next goal is to obtain a quantitative Voronovskaja-type result for the $(α, λ, s)$ -BKSO, $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ),$ with the help of first-order Ditzian-Totik modulus of smoothness (see (Ditzian and Totik Citation1987)). The first-order Ditzian-Totik modulus of smoothness of a function $ξ \in C [0, 1]$ is defined as

ω_{Φ}^{1} (ξ, t) := sup_{0 < | ζ | \leq t} {∥Δ_{ζ Φ (\cdot)}^{1} ξ (\cdot)∥}_{\infty},

where $Φ (τ) = \sqrt{τ (1 - τ)}$ , $τ \in [0, 1]$ and

Δ_{ζ Φ (τ)}^{1} ξ (τ) = \{\begin{matrix} ξ (τ + \frac{1}{2} ζ Φ (τ)) - ξ (τ - \frac{1}{2} ζ Φ (τ)), & if τ \pm \frac{ζ Φ (τ)}{2} \in 0, 1], \\ 0, & otherwise . \end{matrix}

Let $A C_{loc} [0, 1]$ denote the class of absolutely continuous functions defined on $[a, b] \subset 0, 1] .$ Also, for any $\tilde{ξ} \in C [0, 1],$ let $W_{Φ}^{1} [0, 1]$ be the set of all such $\tilde{ξ}$ satisfying $\tilde{ξ} \in A C_{loc} [0, 1]$ and ${∥Φ {\tilde{ξ}}^{'}∥}_{\infty} < \infty$ . Then, we define the corresponding $K$ -functional of $ω_{Φ}^{1} (ξ, t)$ by

K_{Φ}^{1} (ξ, t) = inf \{{∥ξ - \tilde{ξ}∥}_{\infty} + t {∥Φ {\tilde{ξ}}^{'}∥}_{\infty} : \tilde{ξ} \in W_{Φ}^{1} [0, 1], t > 0\} .

In (Ditzian and Totik Citation1987), Ditzian and Totik proved that

(43)

L^{- 1} ω_{Φ}^{1} (ξ, t) \leq K_{ϕ}^{1} (ξ, t) \leq L ω_{Φ}^{1} (ξ, t), 0 < t \leq t_{0},

(43)

for some constants $t_{0}$ and $L > 0.$ We hall note that $L$ is independent of $τ$ and $μ$ .

Theorem 7.

Let $ξ \in C^{2} [0, 1]$ and constant $L$ be defined as in (4.8). Moreover, define the terms $γ_{μ} (τ)$ and $θ_{μ} (τ)$ as

γ_{μ} (τ) = S_{μ, σ, ρ}^{(α, λ, s)} ((t - τ); τ), θ_{μ} (τ) = S_{μ, σ, ρ}^{(α, λ, s)} ((t - τ)^{2}; τ) .

Then for all $τ \in 0, 1]$ and $μ \geq s$ , we have

|S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ) - ξ (τ) - γ_{μ} (τ) ξ^{'} (τ) - \frac{θ_{μ} (τ)}{2} ξ^{^{''}} (τ)| \leq \frac{4 L}{μ} τ (1 - τ) ω_{Φ}^{1} (ξ^{^{''}}, \frac{\sqrt{6}}{\sqrt{n}}) .

Proof.

For $τ, t \in 0, 1]$ consider the function $ξ \in C^{2} [0, 1]$ . If we apply Taylor’s formula to function $ξ$ $\in C [0, 1]$ , we obtain

ξ (t) = ξ (τ) + (t - τ) ξ^{'} (τ) + \int_{τ}^{t} (t - y) ξ^{^{''}} (y) dy .

Subsequently, we can write

(44)

ξ (t) - ξ (τ) - (t - τ) ξ^{'} (τ) - \frac{ξ^{^{''}} (τ)}{2} (t - τ)^{2} = \int_{τ}^{t} (t - y) [ξ^{^{''}} (y) - ξ^{^{''}} (τ)] dy .

(44)

Next, utilization of $(α, λ, s)$ -BKSO to both sides of (4.9) yields

(45)

|S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ) - ξ (τ) - γ_{μ} (τ) ξ^{'} (τ) - θ_{μ} (τ) \frac{ξ^{^{''}} (τ)}{2}| \leq S_{μ, σ, ρ}^{(α, λ, s)} (|\int_{τ}^{t} |t - y| |ξ^{^{''}} (y) - ξ^{^{''}} (τ)| dy|; τ) .

(45)

Inspired by (Finta Citation2011), we can estimate the quantity on the right-hand side of (4.10) as

|\int_{τ}^{t} |t - y| |ξ^{^{''}} (y) - ξ^{^{''}} (τ)| dy| \leq 2 ∥ ξ^{^{''}} - \tilde{ξ} ∥_{\infty} (t - τ)^{2} + 2 {∥Φ {\tilde{ξ}}^{'}∥}_{\infty} Φ^{- 1} (τ) {|t - τ|}^{3},

where $\tilde{ξ} \in W_{Φ}^{1} [0, 1]$ . Then utilizing identities (3.16) and (3.17) from Corollary 2 for $μ \geq s$ , we get

(46)

S_{μ, σ, ρ}^{(α, λ, s)} ((t - τ)^{2}; τ) \leq \frac{2}{μ} Φ^{2} (τ) and S_{μ, σ, ρ}^{(α, λ, s)} ((t - τ)^{4}; τ) \leq \frac{12}{μ^{2}} Φ^{4} (τ) .

(46)

Consequently, we conclude

|S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ) - ξ (τ) - γ_{μ} (τ) ξ^{'} (τ) - θ_{μ} (τ) \frac{ξ^{^{''}} (τ)}{2}|

\leq 2 {∥ξ - \tilde{ξ}∥}_{\infty}_{μ, σ, ρ}^{(α, λ, s)} ({(t - τ)}^{2}; τ) + 2 {∥Φ {\tilde{ξ}}^{'}∥}_{\infty} Φ^{- 1} (τ)_{μ, σ, ρ}^{(α, λ, s)} (| t - τ |^{3}; τ)

\leq \frac{4}{μ} Φ^{2} (τ) {∥ξ^{^{''}} - \tilde{ξ}∥}_{\infty} + 2 {∥Φ {\tilde{ξ}}^{'}∥}_{\infty} {\tilde{ξ}}^{- 1} (τ) {[S_{μ, σ, ρ}^{(α, λ, s)} ((t - τ)^{2}; τ)]}^{1 / 2} {[S_{μ, σ, ρ}^{(α, λ, s)} ((t - τ)^{4}; τ)]}^{1 / 2}

(47)

\leq \frac{4}{μ} Φ^{2} (τ) [{∥ξ^{^{''}} - \tilde{ξ}∥}_{\infty} + \frac{\sqrt{6}}{\sqrt{μ}} {∥Φ {\tilde{ξ}}^{'}∥}_{\infty}],

(47)

by combining inequalities (4.10)–(4.11) and using the Cauchy-Schwarz inequality. Lastly, we get

|S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ) - ξ (τ) - γ_{μ} (τ) ξ^{'} (τ) - \frac{θ_{μ} (τ)}{2} ξ^{^{''}} (τ)| \leq \frac{4 L}{μ} Φ^{2} (τ) ω_{Φ}^{1} (ξ^{^{''}}, \frac{\sqrt{6}}{\sqrt{μ}}),

by taking the infimum of the right-hand side of (4.12) over all $\tilde{ξ} \in W_{Φ}^{1} [0, 1] .$ □

The next corollary directly results from Theorem 7.

Corollary 4.

For $B \in ℜ^{+}$ and $ξ \in C^{2} [0, 1]$ , we have

[sta t_{B}, η_{μ}] - lim_{μ \to \infty} μ [S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ) - ξ (τ) - γ_{μ} (τ) ξ^{'} (τ) - \frac{θ_{μ} (τ)}{2} ξ^{^{''}} (τ)] = 0.

5. Conclusions with numerical results

In our manuscript, we introduced the operators $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ with the inclusion of the shape parameters $λ \in [- 1, 1],$ $α \in [0, 1]$ and a positive parameter $s$ . We presented a uniform convergence result and analysed the convergence properties of $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ . In addition, we gave a Voronovskaja-type approximation result with the help of the weighted $B -$ statistical convergence notion and estimated the rate of the weighted $B -$ statistical convergence of these operators.

Now, to observe the approximation behaviour of the operators $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ),$ we consider the function $ξ (τ) = τ^{3} sin (πτ) cos (πτ) .$ From , we can easily say that $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ behaves consistently with the given function $ξ (τ),$ and we observe that approximation behaviour of $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ shows improvement since an error in the calculations decreases as $μ$ value increases (see ). Furthermore, presents the absolute errors for $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ , the $α -$ Bernstein operator $T_{μ, α} (ξ; τ)$ given in (2.1), the Stancu variant of $α -$ Bernstein-Kantorovich operators $S_{μ, α}^{σ, ρ} (ξ; τ)$ constructed in (Mohiuddine and Özger Citation2020) and the Kantorovich variant of modified Bernstein operators $K_{μ}^{M, 1} (ξ; τ)$ introduced in (Gupta et al. Citation2019), respectively. It can be deduced from that $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ gives a better approximation for the given function $ξ (τ)$ compared to the operators $T_{μ, α} (ξ; τ),$ $S_{μ, α}^{σ, ρ} (ξ; τ)$ and $K_{μ}^{M, 1} (ξ; τ) .$ Due to the inclusion of shape parameters $λ,$ $α$ and the positive parameter $s$ together within the basis function, it is clear to see that the operators $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ are a generalization of the other operators from the literature, which are listed in . It is imperative to note that it is possible to enhance the approximation behaviour of operators $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ by altering their bases functions or taking the linear and repetitive combinations of the mentioned operators. For example, the $q$ modification of these operators can be examined as future work.

Figure 1. Approximations by $S_{μ, 1, 1}^{(0.9, - 1, 2)} (τ^{3} sin (πτ) cos (πτ); τ)$ with specific $μ$ values.

Figure 2. Errors of the approximation of $S_{μ, 1, 1}^{(0.9, - 1, 2)} (τ^{3} sin (πτ) cos (πτ); τ)$ with specific $μ$ values.

Table 1. Absolute errors for the operators $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ , $T_{μ, α} (ξ; τ)$ , $S_{μ, α}^{σ, ρ} (ξ; τ)$ , and $K_{μ}^{M, 1} (ξ; τ)$ for $ξ (τ) = τ^{3} sin (πτ) cos (πτ)$ when $λ = 0.5$ , $α = 0.9$ , $σ = 1$ and $ρ = 1$ .

Display Table

Table 2. Reduction of the operators $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ for specific values of the various parameters.

Display Table

Acknowledgments

The authors have greatly benefitted from the constructive feedback of the editor and anonymous referees. Therefore, they express their gratitude for the valuable comments provided on the initial draft of this paper, which contributed to enhancing its presentation and readability.

Disclosure statement

No potential conflict of interest was reported by the author(s).

References

Acu AM, Manav N, Sofonea DF. 2018. Approximation properties of λ-Kantorovich operators. J Inequal Appl. 2018(1):Article ID 202. doi: 10.1186/s13660-018-1795-7.
Google Scholar
Aktuğlu H, Gezer H, Baytunç E, Atamert MS. 2022. Approximation properties of generalized blending type lototsky-Bernstein operators. J Math Inequal. 16(2):707–728. doi: 10.7153/jmi-2022-16-50.
Web of Science ®Google Scholar
Ansari KJ, Karakılıç S, Özger F. 2022a. Bivariate Bernstein-kantorovich operators with a summability method and related GBS operators. Filomat. 36(19):6751–6765. doi: 10.2298/FIL2219751A.
Web of Science ®Google Scholar
Ansari KJ, Özger F, Ödemiş Özger Z. 2022b. Numerical and theoretical approximation results for schurer-stancu operators with shape parameter λ. Comp Appl Math. 41(4):1–18. doi: 10.1007/s40314-022-01877-4.
Google Scholar
Aslan R. 2022. On a stancu form szász-mirakjan-kantorovich operators based on shape parameter λ. Adv Studies: Euro-Tbilisi Math J. 15(1):151–166. doi: 10.32513/asetmj/19322008210.
Google Scholar
Aslan R. 2023. Approximation properties of univariate and bivariate new class $\lambda $-Bernstein–Kantorovich operators and its associated GBS operators. Comp Appl Math. 42(1):Article ID 34. DOI. doi: 10.1007/s40314-022-02182-w.
Google Scholar
Aslan R. 2024. Rate of approximation of blending type modified univariate and bivariate λ-Schurer-Kantorovich operators. Kuwait J Sci. 51(1):100168. doi: 10.1016/j.kjs.2023.12.007.
Web of Science ®Google Scholar
Ayman-Mursaleen M, Nasiruzzaman M, Rao N, Dilshad M, Nisar KS. 2024. Approximation by the modified $ \lambda $-Bernstein-polynomial in terms of basis function. Math. 9(2):4409–4426. doi: 10.3934/math.2024217.
Google Scholar
Ayman-Mursaleen M, Rao N, Rani M, Kilicman A, Al-Abied AA, Malik P, Gobithaasan RU. 2023. A Note on Approximation of Blending Type Bernstein–Schurer–Kantorovich Operators with Shape Parameter α. J Math. 2023:1–13. doi: 10.1155/2023/5245806.
Web of Science ®Google Scholar
Baliarsingh P, Kadak U, Mursaleen M. 2018. On statistical convergence of difference sequences of fractional order and related Korovkin type approximation theorems. Quaest Math. 41(8):1117–1133. doi: 10.2989/16073606.2017.1420705.
Web of Science ®Google Scholar
Bărbosu D. 2004. Kantorovich-Stancu type operators. J Inequal Pure Appl Math. 5(3):1–6. doi: 10.18514/MMN.2004.71.
Google Scholar
Belen C, Mohiuddine SA. 2013. Generalized weighted statistical convergence and application. Appl Math Comput. 219(18):9821–9826. doi: 10.1016/j.amc.2013.03.115.
Web of Science ®Google Scholar
Bell HT. 1973. Order summability and almost convergence. Proc Amer Math Soc. 38(3):548–552. doi: 10.1090/S0002-9939-1973-0310489-8.
Web of Science ®Google Scholar
Bernstein S. 1912. Dámonstration du tháorème de Weierstrass fondeá sur le calcul des probabilitás. Commun Soc Math Kharkow. 13(1):1–2.
Google Scholar
Bodur M, Manav N, Tasdelen F. 2022. Approximation properties of λ-Bernstein-kantorovich-stancu operators. Math Slovaca. 72(1):141–152. doi: 10.1515/ms-2022-0010.
Web of Science ®Google Scholar
Bohman H. 1952. On approximation of continuous and of analytic functions. Ark Mat. 2(1):43–56. doi: 10.1007/BF02591381.
Google Scholar
Bustamante J. 2017. Bernstein operators and their properties. Cham: Springer. doi:10.1007/978-3-319-55402-0.
Google Scholar
Cai Q, Ansari KJ, Temizer Ersoy M, Özger F. 2022. Statistical Blending-Type Approximation by a Class of Operators That Includes Shape Parameters λ and α. Math. 10(7):1149. doi: 10.3390/math10071149.
Google Scholar
Cai QB, Lian BY, Zhou G. 2018. Approximation properties of λ-Bernstein operators. J Inequal Appl. 2018(61):1–11. doi: 10.1186/s13660-018-1653-7.
PubMedGoogle Scholar
Chen M-Y, Nasiruzzaman M, Mursaleen MA, Rao N, Kiliçman A. 2022. On shape parameter α based approximation properties and q-statistical convergence of baskakov-gamma operators. Jour Mathematics. 2022:4190732. https://www.hindawi.com/journals/jmath/2022/4190732.
Web of Science ®Google Scholar
Chen X, Tan J, Liu Z, Xie J. 2017. Approximation of functions by a new family of generalized Bernstein operators. J Math Anal Appl. 450(1):244–261. doi: 10.1016/j.jmaa.2016.12.075.
Web of Science ®Google Scholar
Ditzian Z, Totik V. 1987. Moduli of Smoothness. In: Springer series in computational mathematics. Vol. 9, New York, NY: Springer-Verlag. doi: 10.1007/978-1-4612-4778-4.
Google Scholar
Finta Z. 2011. Remark on Voronovskaja theorem for q-Bernstein operators. Stud Univ Babeş-Bolyai Math. 56(2):335–339.
Google Scholar
Gezer H, Aktuğlu H, Baytunç E, Atamert MS. 2022. Generalized blending type Bernstein operators based on the shape parameter λ. J Inequal Appl. 2022(96):1–19. doi: 10.1186/s13660-022-02832-x.
Google Scholar
Gupta V, Tachev G, Acu AM. 2019. Modified Kantorovich operators with better approximation properties. Numer Algor. 81(1):125–149. doi: 10.1007/s11075-018-0538-7.
Web of Science ®Google Scholar
Kadak U. 2018. Modularly weighted four dimensional matrix summability with application to Korovkin type approximation theorem. J Math Anal Appl. 468(1):38–63. doi: 10.1016/j.jmaa.2018.06.047.
Web of Science ®Google Scholar
Kadak U. 2020. Statistical summability of double sequences by the weighted mean and associated approximation results. In: Dutta H, Peters J, editors. Applied mathematical analysis: theory, methods, and applications. Cham: Springer; pp. 61–85.
Google Scholar
Kadak U, Braha NL, Srivastava HM. 2017. Statistical weighted B-summability and its applications to approximation theorems. Appl Math Comput. 302:80–96. doi: 10.1016/j.amc.2017.01.011.
Web of Science ®Google Scholar
Kadak U, Özger F. 2021. A numerical comparative study of generalized Bernstein-kantorovich operators. Math Found Comput. 4(4):311–332. doi: 10.3934/mfc.2021021.
Web of Science ®Google Scholar
Kantorovich LV. 1930. Sur certains developpements suivant les polynômes de la forme de S.Bernstein I, II. Dokl Akad Nauk SSSR. 563–568, 595–600.
Google Scholar
Kolk E. 1993. Matrix summability of statistically convergent sequences. Analysis. 13(1–2):77–83. doi: 10.1524/anly.1993.13.12.77.
Google Scholar
Korovkin PP. 1960. Linear operators and approximation theory. Delhi, India: Hindustan Publishing Corporation.
Google Scholar
Milovanović GV, Mursaleen M, Nasiruzzaman M. 2018. Modified Stancu type dunkl generalization of szász–Kantorovich operators. RACSAM. 112(1):135–151. doi: 10.1007/s13398-016-0369-0.
Google Scholar
Mohiuddine SA. 2016. Statistical weighted A-summability with application to Korovkin’s type approximation theorem. J Inequal Appl. 2016(1):Article ID 101. doi: 10.1186/s13660-016-1040-1.
Google Scholar
Mohiuddine SA, Acar T, Alotaibi A. 2017. Construction of a new family of Bernstein-Kantorovich operators. Math Meth Appl Sci. 40(18):7749–7759. doi: 10.1002/mma.4559.
Web of Science ®Google Scholar
Mohiuddine SA, Özger F. 2020. Approximation of functions by Stancu variant of Bernstein–Kantorovich operators based on shape parameter ${\varvec{\alpha}}$. RACSAM. 114(2):Article ID 70. doi: 10.1007/s13398-020-00802-w.
Google Scholar
Mursaleen M, Karakaya V, Ertürk M, Gürsoy F. 2012. Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl Math Comput. 218(18):9132–9137. doi: 10.1016/j.amc.2012.02.068.
Web of Science ®Google Scholar
Nasiruzzaman M, Rao N, Wazir S, Kumar R. 2019. Approximation on parametric extension of baskakov–durrmeyer operators on weighted spaces. J Inequal Appl. 2019(1):Article ID 103. doi: 10.1186/s13660-019-2055-1.
Google Scholar
Özger F. 2019. Weighted statistical approximation properties of univariate and bivariate λ-Kantorovich operators. Filomat. 33(11):3473–3486. doi: 10.2298/FIL1911473O.
Web of Science ®Google Scholar
Özger F. 2020. Applications of Generalized Weighted Statistical Convergence to Approximation Theorems for Functions of One and Two Variables. Numer Funct Anal Optim. 41(16):1990–2006. doi: 10.1080/01630563.2020.1868503.
Web of Science ®Google Scholar
Özger F, Aljimi E, Temizer Ersoy M. 2022. Rate of weighted statistical convergence for generalized blending-type Bernstein-kantorovich operators. Mathematics. 10(12):Article ID 2027. doi: 10.3390/math10122027.
Web of Science ®Google Scholar
Özger F, Demirci K, Yıldız S. 2020. Approximation by Kantorovich variant of λ-schurer operators and related numerical results. In: Dutta H, editor. Topics in contemporary mathematical analysis and applications. Boca Raton: CRC Press; p. 77–94. ISBN 9780367532666.
Google Scholar
Savaş E, Mursaleen M. 2023. Bézier Type Kantorovich q-Baskakov Operators via Wavelets and Some Approximation Properties. Bull Iran Math Soc. 49(5). doi: 10.1007/s41980-023-00815-2.
Web of Science ®Google Scholar
Srivastava HM, Ansari KJ, Özger F, Ödemiş Özger Z. 2021. A Link between Approximation Theory and Summability Methods via Four-Dimensional Infinite Matrices. Math. 9(16):1895. doi: 10.3390/math9161895.
Google Scholar
Stancu DD. 1968. Approximation of function by a new class of polynomial operators. Rev Roum Math Pures Appl. 13(8):1173–1194.
Google Scholar
Stieglitz M. 1973. Eine verallgemeinerung des begriffs der fastkonvergenz. Math Japon. 18(1):53–70.
Google Scholar
Weierstrass VK. 1885. Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen. Sitzungsberichte der Königlich Preuß ischen Akademie der Wissenschaften zu Berlin. 2:633–639.
Google Scholar
Ye Z, Long X, Zeng XM, Adjustment algoritheorems for bézier curve and surface, in: The 5. International Conference on Computer Science and Education, 2010, 1712–1716. doi: 10.1109/ICCSE.2010.5593563
Google Scholar

Kantorovich-Stancu type (α,λ,s) - Bernstein operators and their approximation properties

ABSTRACT

1. Introduction

2. Essential preliminaries

3. New (α, λ, s) – Bernstein-Kantorovich-Stancu operators of blending-type

4. Convergence Results for Kantorovich-Stancu Type (α, λ, s)-Bernstein Operators $S_{μ, σ, ρ}^{(α, λ, s)}$

5. Conclusions with numerical results

Table 1. Absolute errors for the operators $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ , $T_{μ, α} (ξ; τ)$ , $S_{μ, α}^{σ, ρ} (ξ; τ)$ , and $K_{μ}^{M, 1} (ξ; τ)$ for $ξ (τ) = τ^{3} sin (πτ) cos (πτ)$ when $λ = 0.5$ , $α = 0.9$ , $σ = 1$ and $ρ = 1$ .

Table 2. Reduction of the operators $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ for specific values of the various parameters.

Acknowledgments

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References

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Kantorovich-Stancu type (α,λ,s) - Bernstein operators and their approximation properties

ABSTRACT

1. Introduction

2. Essential preliminaries

3. New (α, λ, s) – Bernstein-Kantorovich-Stancu operators of blending-type

4. Convergence Results for Kantorovich-Stancu Type (α, λ, s)-Bernstein Operators Sμ,σ,ρα,λ,s

5. Conclusions with numerical results

Table 1. Absolute errors for the operators Sμ,σ,ρα,λ,s(ξ;τ), Tμ,α(ξ;τ), Sμ,ασ,ρ(ξ;τ), and KμM,1(ξ;τ) for ξτ=τ3sinπτcosπτ when λ=0.5, α=0.9, σ=1 and ρ=1.

Table 2. Reduction of the operators Sμ,σ,ρα,λ,s(ξ;τ) for specific values of the various parameters.

Acknowledgments

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4. Convergence Results for Kantorovich-Stancu Type (α, λ, s)-Bernstein Operators $S_{μ, σ, ρ}^{(α, λ, s)}$

Table 1. Absolute errors for the operators $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ , $T_{μ, α} (ξ; τ)$ , $S_{μ, α}^{σ, ρ} (ξ; τ)$ , and $K_{μ}^{M, 1} (ξ; τ)$ for $ξ (τ) = τ^{3} sin (πτ) cos (πτ)$ when $λ = 0.5$ , $α = 0.9$ , $σ = 1$ and $ρ = 1$ .

Table 2. Reduction of the operators $S_{μ, σ, ρ}^{(α, λ, s)} (ξ; τ)$ for specific values of the various parameters.