In this paper, we define and study q-statistical limit point, q-statistical cluster point, q-statistically Cauchy, q-strongly Cesàro and statistically $C_1^q$-summable sequences. We establish relationships of q-statistical convergence with q-statistically Cauchy, q-strongly Cesàro and statistically $C_1^q$-summable sequences. Further, we apply q-statistical convergence to prove a Korovkin type approximation theorem.

In this article by means of shifted knots properties, we introduce a new type of coupled Bernstein operators for Bézier basis functions. First, we construct the operators based on shifted knots properties of Bézier basis functions then investigate the Korovkin's theorem, establish a local approximation theorem, and provide a convergence theorem for Lipschitz continuous functions and Peetre's $ K $-functional. In addition, we also obtain an asymptotic formula of the type Voronovskaja.

Given a Lebesgue integrable function $f$ over $[-\pi,\pi]$, we consider the sequence of matrices $\{Y_nT_n[f]\}_n$, where $T_n[f]$ is the $n$-by-$n$ Toeplitz matrix generated by $f$ and $Y_n$ is th...

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 .