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 .

Our main purpose of this article is to study the convergence and other related properties of q-Bernstein-Kantorovich operators including the shifted knots of real positive numbers. We design the shifted knots of Bernstein-Kantorovich operators generated by the basic q-calculus. More precisely, we study the convergence properties of our new operators in the space of continuous functions and Lebesgue space. We obtain the degree of convergence with the help of modulus of continuity and integral modulus of continuity. Furthermore, we establish the quantitative estimates of Voronovskaja-type.