Approximation by Szász-Jakimovski-Leviatan-Type Operators via Aid of Appell Polynomials

The main purpose of the present article is to construct a newly Szász-Jakimovski-Leviatan-type positive linear operators in the Dunkl analogue by the aid of Appell polynomials. In order to investigate the approximation properties of these operators, first we estimate the moments and obtain the basic results. Further, we study the approximation by the use of modulus of continuity in the spaces of the Lipschitz functions, Peetres K-functional, and weighted modulus of continuity. Moreover, we study A -statistical convergence of operators and approximation properties of the bivariate case.


Global Approximation
In the present section, we follow Gadžiev [11] and recall the weighted spaces of the functions on ½0, ∞Þ, as well as additional conditions under which the analogous theorem of P.P. Korovkin holds for such a kind of functions. Take y ⟶ ϕðyÞ be continuous and strictly increasing function with σðyÞ = 1 + ϕ 2 ðyÞ and lim y→∞ σðyÞ = ∞. Let B σ ½0, ∞Þ be a set of functions defined on [0, ∞), verifying the results where K h is a constant and depending only on function h and B σ ½0, ∞Þ is space of all continuous as well as bounded functions on ½0, ∞Þ. Let the set of all continuous functions on ½0 , ∞Þ will be denoted by C σ ½0, ∞Þ and B σ ½0,∞Þ ⊂ C σ ½0,∞Þ equipped with the norm khk σ = sup y∈½0,∞Þ jhj/σðyÞ.

Let us denote
It is well known that (see [26]) the sequence of linear positive operators fL n g n≥1 maps C σ ½0, ∞Þ into B σ ½0, ∞Þ if and only if 3 Journal of Function Spaces where C is a positive constant.
Proof. We prove this theorem by applying Korovkin's theorem so it is sufficient to show that From Lemma 2, we easily see that Similarly, for which imply that kJ * n,λ ðϕ 1 ; which clearly shows that kJ * n,λ ðϕ 2 ; yÞ − y 2 k σ ⟶ 0, whenever n ⟶ ∞: For all h ∈ C B ½0, ∞Þ, operators given by (5) satisfy where δ n ðyÞ = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi J * n,λ ðψ 2 ; yÞ q and C B ½0, ∞Þ stand for space of all continuous and bounded functions defined on ½0, ∞Þ.
Proof. We prove Theorem 7 by using the well-known Cauchy-Schwarz inequality and modulus of continuity. Thus, we see that If we take δ = δ n = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi J * n,λ ðψ 2 ; yÞ q , we get the required result asserted by Theorem 7.

Some Direct Results of J * n,λ
The present section gives some direct approximation results in the space of K-functional and in the Lipschitz spaces. We suppose the following.
Definition 8. For every δ > 0 and h ∈ C½0,∞Þ, we define Journal of Function Spaces where C 2 B ½0, ∞Þ is defined by Now, there exists an absolute constant C > 0 such that where ω 2 ðh ; δÞ is the second-order modulus of continuity given by Moreover, the modulus of continuity of order one is Theorem 9. Let h ∈ C 2 B ½0, ∞Þ, we define an auxiliary operators K * n,λ such that Then, for every ψ ∈ C 2 B ½0, ∞Þ, operators K * n,λ satisfy where Θ n ðyÞ = ðδ n ðyÞÞ 2 + ð1/n 2 ÞððF′ð1Þ/Fð1ÞÞ + 2λÞ 2 and δ n ðyÞ are defined in Theorem 7.
Proof. Take ψ ∈ C 2 B ½0, ∞Þ; then, we easily conclude that K * n,λ ðϕ 0 ; yÞ = 1 and We also know easily Therefore, From the Taylor series we see Applying K * n,λ , we have Since we know Therefore, we get This gives the complete proof.

Journal of Function Spaces
Theorem 10. Let h ∈ C B ½0, ∞Þ and any ψ ∈ C 2 B ½0, ∞Þ. Then, there exists a constant C > 0 such that where Θ n ðyÞ is defined by Theorem 9.
Proof. To prove Theorem 11, we use the well-known Hölder inequality by applying (43) The proof is complete.
From [28] for an arbitrary h ∈ C m σ ½0,∞Þ, the weighted modulus of continuity is introduced such that The two main properties of this modulus of continuity are lim δ→0 Ωðh ; δÞ = 0 and where s, y ∈ ½0,∞Þ.

A-Statistical Convergence
Here, we obtain the A-statistical convergence for the operators J * n,λ by (5). From [29], we recall the needed notations and notions for A-statistical convergence. Take G = ðD nk Þ be a nonnegative infinite summability matrix. For a given sequence y = ðy k Þ, the A-transform of y is denoted by Gy : ðGyÞ n where the series converges for each n and defined by The matrix G is said to be regular if limðGyÞ n = L whenever limx = L and y = ðy n Þ are said to be a A-statis-tically convergent to L, i.e., st G − limy = L if for each ∈>0 , lim n ∑ k:jy k −Lj≥∈ D nk = 0. For the recent work on statistical convergence and statistical approximation, we refer to [30][31][32][33][34][35][36][37].
Theorem 13. Let operators J * n,λ be defined by 1 and a nonnegative regular summability matrix be G = ðD nk Þ; then, for every h ∈ C m σ ½0, ∞Þ Proof. It is enough to show that From Lemma 2, we conclude that which implies that Similarly for j = 2 Journal of Function Spaces which shows that For a given ε > 0, we define the sets such that Therefore, we conclude that This is denumerable to complete the proof.