A Link between Approximation Theory and Summability Methods via Four-Dimensional Inﬁnite Matrices

: In this study, we present a link between approximation theory and summability methods by constructing bivariate Bernstein-Kantorovich type operators on an extended domain with reparametrized knots. We use a statistical convergence type and power series method to obtain certain Korovkin type theorems, and we study certain rates of convergences related to these summability methods. Furthermore, we numerically analyze the theoretical results and provide some computer graphics to emphasize the importance of this study.


Introduction
The most well known proof of Weierstrass approximation theorem (see [1]) was given in [2,3]. Bernstein opened a new way by constructing a sequence of polynomials depending explicitly on evaluation of a function at rational values. Researchers have successfully extended this idea for approximating functions, for instance, L.V. Kantorovich introduced a new process to approximate Lebesgue integrable real-valued functions defined on [0, 1] (see [4]). Recently, there has been an increasing degree of attention on approximation properties of Bernstein type operators with shape parameters (see [5][6][7][8][9][10][11][12]).
The decision on whether a sequence of positive linear operators converges strongly includes the use of Korovkin-type theorems. Using certain types of statistical convergences instead of usual convergence in Korovkin type approximation theory provides several benefits. The statistical convergence extends the scope of classical convergence of sequences of numbers or functions, and it has been used in various fields of mathematics such as summability theory [13], topology [14], optimization [15], measure theory [16], number theory [17], trigonometric series [18], approximation by positive linear operators [9,[19][20][21][22][23][24][25]. Statistical convergence of double and single sequences were given in [26][27][28]. Unlike any convergent sequence, statistically convergent double or single sequences do not need to be bounded. This is why it is preferred to be used by many researchers in approximation theory (see, for instance, [29][30][31]).
The primary objective of this work is to establish a link between approximation theory and summability methods via four-dimensional matrices and construction of bivariate Bernstein-Kantorovich type operators on extended domain with reparametrized knots, as well as to prove some Korovkin theorems using two summability methods motivated by the studies [32][33][34][35][36]. The first summability method is a statistical convergence concept which is stronger than the classical case and the second one is power series method (PSM). Since we create a link between the approximation theory and the summability theory we obtain the rate of convergence for PSM and the rate of statistical convergence by modulus of continuity (MC). Moreover, we provide some computer graphics to numerically analyze the efficiency and the accuracy of convergence of our operators, and obtain corresponding error and density plots. Finally, we provide some concluding remarks to emphasize main concepts of this article. All the results that have been obtained in the present paper can be extended for n-variate functions.

Auxiliary Results
Certain notions and auxiliary results are given in this section. Let = ( r,s ) be a double sequence of real numbers. Assume that there is N = N(τ) ∈ N for each τ > 0, so that | r,s − Q| < τ whenever r, s > N, in this case double sequence = ( r,s ) is said to be convergent to Q in Pringsheim's sense (or simply Π-convergent), and it is denoted by Π − lim r,s r,s = Q, where Q is a real number (see [37]). When there is a positive number E such that | r,s | ≤ E for all (r, s) ∈ N 2 = N × N, the double sequence is said to be bounded. As it is well known, every convergent single sequence is bounded whereas a convergent double sequence need not to be bounded.
Assume that D = (d l,o,r,s ) is a four-dimensional summability method. Given a double sequence = ( r,s ), D transform of , denoted by D := ((D ) l,o ), is defined by d l,o,r,s r,s , and the double series is Π-convergent for (l, o) ∈ N 2 . When a four-dimensional matrix D = (d l,o,r,s ) maps every bounded Π−convergent sequence into a Π−convergent sequence with the same Π−limit, it is called RH−regular (shortly RHR). A four-dimensional matrix r,s < E 1 is satisfied for finite positive integers E 1 and E 2 and for each (l, o) ∈ N 2 . These conditions are called Robison-Hamilton conditions [38]. Assume that D = (d l,o,r,s ) is a nonnegative RHR matrix, and S ⊂ N 2 , then the D−density of S is defined by provided that the limit on the right-hand side exists in the Pringsheim sense. A real double sequence = ( r,s ) is called D−statistically convergent to Q and denoted by st 2 D − lim r,s r,s = Q if, for every τ > 0, (see also [31,39]). A Π−convergent double sequence is D−statistically convergent to the same number even if the converse statement is not true. When the D = C(1, 1), C(1, 1)−statistical convergence becomes statistical convergence for double sequences (see also [27]), where C(1, 1) = (c l,o,r,s ) is the double Cesàro matrix, defined by c l,o,r,s = 1/lo if 1 ≤ r ≤ o, 1 ≤ s ≤ l, and c l,o,r,s = 0 otherwise.
Suppose that (ξ r,s ) is a double sequence of nonnegative numbers with condition ξ 0,0 > 0, then the power series are satisfied for any µ, υ [40].
In this work, we assume that PSM is regular.

Statistical Convergence via Four Dimensional Matrices
A bivariate case of the γ Kantorovich operators, defined in [41], is constructed in this section. Moreover, the D-statistical convergence of these bivariate operators is studied.
The parametric extensions of (2) and (3) for r, s ∈ N and h ∈C κ,β are the operators where Lemma 1. The parametric extensions of operators defined in (5) and (6) are linear and positive.
Proof. The assertion follows from the definitions of K r,s,κ,β :C α,β −→C defined for any r, s ∈ N and any h ∈C κ,β by the relation Proof. We get the desired result by direct computation, taking into account the definitions (5), (6) and Lemma 1. Proof. Using the fact that product of linear and positive operators are also linear and positive, and applying Lemma 1 we obtain desired result.
In the recent paper [41], the following results were provided: and κ be a non-negative integer, then the moments of Bernstein-Kantorovich type operators on extended domain with reparametrized knots are as follows: Lemma 5. The parametric extension K γ 1 ,y r,κ satisfies the identities (8), (9) and (10).
The following lemmas are stated to give moments. Lemma 6. Let e uv = s u t v , u, v ∈ N, y, z ∈ R be the two-dimensional test functions. The bivariate operators defined in (7) satisfy Proof. Taking into account definition (7) and Lemma 5, the result follows.
By applying Lemma 6, we get the relation (11). Similarly we have the equality (12).
Theorem 1 provides next result.
Proof. We now claim that Following result is satisfied by Lemma 6 (a): This result guarantees that (13) holds for u = 0.
Defining the sets and one can obtain Similarly we have that is (13) holds for u = 2. Finally, taking into account the inequalities and defining the sets that is (13) holds for u = 3. As a result, K γ 1 ,γ 2 r,s,κ,β satisfies all hypothesis of Theorem 1 which concludes the proof.
The following corollary is obtained by replacing the double matrix D in Theorem 1 with the double identity matrix.
The C(1, 1)-statistical convergence becomes statistical convergence for double sequences if D = C(1, 1) is chosen. This leads us to the following corollary: r,s,κ,β via Power Series Method Korovkin type approximation theory by power series method have been studied in several function spaces by many researchers (see [44][45][46][47] ξ r,s a r b s Q r,s (1) C(Ψ) < ∞ (16) throughout this section. Set

The Convergence Rate of Operators
The rate of D-statistical convergence and the rate of convergence for the power series method are calculated in this section with the help of MC. MC is expressed as We know that, for any γ > 0 and for all h ∈ C([o, w] × [e, i]), where [γ] is greatest integer less than or equal to γ (see [48]). The next theorem provides a rate of D-statistical convergence for the proposed operators.
Then taking the supremum over (y, z) ∈ [0, 1] × [0, 1], we have Taking ρ = ρ r,s := 3 we get for any positive integers r, s that Therefore for any τ > 0 we have and from the hypothesis it follows that Next theorem provides a rate of convergence for PSM.

Numerical Results
Final section of this work provides certain numerical experiments and computer graphs supporting the theoretical results. We consider two functions for which we study approximations of our bivariate operators K γ 1 ,γ 2 r,s,κ,β with them and obtain corresponding errors of approximations for different γ 1 , γ 2 , κ, β, r and s values. point represents the evaluation of the plotted function at that point. In Figures 1 and 2, larger values are shown with lighter color. In Figures 1D-F and 2D-F, the color of each point represents the evaluation of the plotted function at that point. In Figures 1 and 2, larger values are shown with lighter color. Example 1. We first consider the function h 1 (y, z) = cos(y 3 ) cos(z 2 ) on (y, z) ∈ [0, 1] × [0, 1]. Choosing γ 1 = γ 2 = 1, κ = β = 2 we obtain some graphs to see the accuracy of the approximations for the function h 1 (y, z). In Figure 3, we give three graphs; the yellow one is the graph of function h 1 (y, z), the blue one is the graph for approximation of our operators when r = s = 20, and finally the green one is also the graph for approximation of our operators when r = s = 60. We present the graph of function h 1 (y, z) and approximations for r = s = 20 and r = s = 60 in Figure 1A-C, respectively. In Figure 1D-F, we give the graphs of corresponding density plots, for instance, (D) is the density plot of (A). In density plots (D)-(F), we represent the color of each corresponding point of the function. Finally we give the errors of approximations of our operators for r = s = 20 and r = s = 60 in Figure 4. It can be seen that the error decreases when the values of r and s increase, as it is expected.

Example 2.
We now consider the function We take γ 1 = γ 2 = −0.5, κ = β = 3 to study approximation of operators K γ 1 ,γ 2 r,s,κ,β for the function h 2 (y, z). We provide graph of h 2 (y, z), graph for approximation of our operators when r = s = 20, and graph for approximation of our operators when r = s = 60 in Figure 5, and we use the colors yellow, blue, and green, respectively. We give the graph of function h 1 (y, z) and approximations for r = s = 20 and r = s = 60 in Figure 2A-C, respectively. In Figure 2D-F, we give the corresponding density plots. Moreover, we obtain the errors of approximations when r = s = 20 and r = s = 60 in Figure 6.   As a result, we show that the operators defined in this paper approximate different kind of functions for certain γ 1 , γ 2 , κ, β, r and s values.

Concluding Remarks
Many mathematicians have investigated the Korovkin-type approximation theorems for a sequence of positive linear operators by different types of convergences. In this study, we focus on two summability methods including double sequences to prove Korovkin type theorems for the proposed operators. We also prove certain rates of convergence theorems connected with these two summability methods and support our theoretical results with numerical experiments. This is why the content of this paper is absolutely different from other types of papers in the literature like [9]. We also note that we reparametrize knots of operators defined in [9] and extend domain of the functions. Now, we show that our results related to power series method are non-trivial generalization of the classical Korovkin results. Using the double sequence r,s = 1 + (−1) r+s , we consider the following operators: U γ 1 ,γ 2 r,s,κ,β = (1 + (1−) r+s )K γ 1 ,γ 2 r,s,κ,β .
It is clear that the operators U γ 1 ,γ 2 r,s,κ,β do not satisfy Korovkin conditions for functions of two variables since U