Construction of q-Baskakov Operators by Wavelets and Approximation Properties

We use wavelets to define the Kantorovich variant of q -Baskakov type operators, and for 1≤ p< ∞ 1 ≤ p < ∞ , we study the Lₚ L p -approximation. Let ξ ξ be any positive constant and Ψ ₖ(x) Ψ k ( x ) be any continuous derivative function such that ∫ RxˢΨ ₖ(x)dqx=0 ∫ R x s Ψ k ( x ) d q x = 0 where 0≤ s ≤ k,\;k∈ N 0 ≤ s ≤ k , k ∈ N , $$0<q<1$$ 0 < q < 1 $$.$$ . For all Ψ ∈ L∞(R) Ψ ∈ L ∞ ( R ) suppose the following conditions hold: (i) a finite positive ξ ξ exits with the property Ψ ⊂ [0,ξ ], sup Ψ ⊂ [ 0 , ξ ] , (ii) its first k moments vanish: For 1≤ s ≤ k,\;k∈ N 1 ≤ s ≤ k , k ∈ N , we have ∫ RtˢΨ (t)dqt=0 ∫ R t s Ψ ( t ) d q t = 0 and ∫ RΨ (t)dqt=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 <jats:alter