Construction of q-Baskakov Operators by Wavelets and Approximation Properties

Abstract

AbstractWe 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 .