Construction of q-Baskakov Operators by Wavelets and Approximation Properties

Abstract

We use wavelets to define the Kantorovich variant of q-Baskakov type operators, and for $$1\le p< \infty$$ 1 ≤ p < ∞ , we study the $$L_{p}$$ L p -approximation. Let $$\xi$$ ξ be any positive constant and $$\Psi _{k}(x)$$ Ψ k ( x ) be any continuous derivative function such that $$\int _{\mathbb {R}}x^{s}\Psi _{k}(x)\mathrm {d}_{q}x=0$$ ∫ R x s Ψ k ( x ) d q x = 0 where $$0\le s \le k,\;k\in \mathbb {N}$$ 0 ≤ s ≤ k , k ∈ N , $$0<q<1$$ 0 < q < 1 $$.$$ . For all $$\Psi \in L_{\infty }(\mathbb {R})$$ Ψ ∈ L ∞ ( R ) suppose the following conditions hold: (i) a finite positive $$\xi$$ ξ exits with the property $$\sup \Psi \subset [0,\xi ],$$ sup Ψ ⊂ [ 0 , ξ ] , (ii) its first k moments vanish: For $$1\le s \le k,\;k\in \mathbb {N}$$ 1 ≤ s ≤ k , k ∈ N , we have $$\int _{\mathbb {R}}t^{s}\Psi (t)\mathrm {d}_{q}t=0$$ ∫ R t s Ψ ( t ) d q t = 0 and $$\int _{\mathbb {R} }\Psi (t)\mathrm {d}_{q}t=1$$ ∫ R Ψ ( t ) d q t = 1 . Then in the sense of Haar basis for $$0<q<1,$$ 0 < q < 1 , the $$q-$$ q - analogue of Baskakov–Kantorovich type wavelets operators are defined by $$\begin{aligned} \left( \mathcal {S}_{r,s,q}\;g\right) (x)=[r]_{q}\sum_{s =0}^{\infty }q^{s -1}B_{r,s,q}(x)\int _{\mathbb {R}}g\left( t\right) \Psi \left( q^{s-1}[r]_{q}t-[s ]_{q}\right) \mathrm {d}_{q}t. \end{aligned}$$ S r , s , q g ( x ) = [ r ] q ∑ s = 0 ∞ q s - 1 B r , s , q ( x ) ∫ R g t Ψ q s - 1 [ r ] q t - [ s ] q d q t .

Sequence spaces derived by $q_{\lambda}$ operators in $\ell _{p}$ spaces and their geometric properties

“…The q-calculus has several applications in different areas, e.g., summability theory [13,14], approximation theory [2,15,18], integral equations [10], etc. Most recently, the Cesàro q-difference sequence spaces have been studied in [22].…”

Approximation properties by Bézier variant of the Baskakov–Schurer–Szász–Stancu operators

“…Also given is their behavior related to the kind of modulus of continuity and smoothness. For recent developments in this direction, we refer to [1][2][3][4][5][6][7][8].…”

A Note on Approximation of Blending Type Bernstein–Schurer–Kantorovich Operators with Shape Parameter α

“…Mohiuddine et al [6], Acu et al [7], İçöz and Çekim [8,9], and Kajla and Micláus [10,11] constructed new sequences of linear positive operators to investigate the rapidity of convergence and order of approximation in diferent functional spaces in terms of several generating functions. Some other researchers developed many other useful operators [6,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] in the same feld. In the recent past, for g ∈ [0, 1], m ∈ N and α ∈ [−1, 1], Chen et al [31] constructed a sequence of new linear positive operators as…”