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