Statistical approximation by means of operators constructed by \(q\)-Lagrange polynomials (Q2912673)

\(q\)-Lagrange polynomials [\textit{A. Altin}, the author and \textit{F. Taşdelen}, Taiwanese J. Math. 10, No. 5, 1131--1137 (2006; Zbl 1148.33004)] are generated by NEWLINENEWLINE\[NEWLINE\prod_{i=1}^{r}\frac{1}{(x_it;q)_{\alpha_i}}=\sum_{m=0}^{\infty}g_{m,q}^{(\alpha_1,\dotsc,\alpha_r)}\;(x_1,\dotsc,x_r)t^mNEWLINE\]NEWLINE NEWLINEwhere \(|t|<\min\left\{|x_1|^{-1},\dotsc,|x_r|^{-1}\right\}\). In this situation, the above equation yields the following explicit representation NEWLINENEWLINE\[NEWLINEg_{m,q}^{(\alpha_1,\dotsc,\alpha_r)}(x_1,\dotsc,x_r)=\sum_{k_1+\dotsb+k_r=m}\left(\prod_{i=1}^{r}(q^{\alpha_i,q})_{k_i}\frac{x_i^{k_i}}{(q,q)_{k_i}}\right), \tag{2}NEWLINE\]NEWLINE NEWLINEwhere, for a complex number \(q\) with \(|q|<1\) and also for arbitrary parameters \(\lambda\) and \(\mu\), NEWLINE\[NEWLINE(\lambda;q)_\mu:=\frac{(\lambda;q)_\infty}{(\lambda q^\mu;q)_\infty} \text{ and } (\lambda;q)_\infty:=\prod_{k=0}^{\infty}(1-\lambda q^k).NEWLINE\]NEWLINE NEWLINEIf \(\mu=n\in\mathbb{N}_0=\mathbb{N}\cup\{0\}\), then this formula reduces to the following oneNEWLINENEWLINE\[NEWLINE(\lambda;q)_\mu:=\begin{cases} 1 & \text{if } n = 0,\\ (1-\lambda)(1-\lambda q)\dotsm(1-\lambda q^{n-1}) & \text{if } n = 1,2,3,\dots. \end{cases}NEWLINE\]NEWLINE NEWLINEIn this paper, the author studies the approximation properties of positive linear operators constructed by \(q\)-Lagrange polynomials for \(0<q<1\) via the concept of statistical convergence. Let \(u^{(i)}:=\left\{u_n^{(i)}\right\}_{n\in\mathbb{N}}\) \((i=1,2,\dotsc,r)\) be sequences of real numbers such that \(0<u_n^{(i)}<1\) for each \(n\in\mathbb{N}\), and let \(0<q<1\). For simplicity, the author writes \(\beta(r):=(u^{(1)}, u^{(2)},\dotsc,u^{(r)})\). By using (2), the author introduces the following positive linear operators on \(C[0,1]\), the space of all continuous real valued function on \([0,1]\): NEWLINENEWLINE\[NEWLINE\begin{multlined} L_n^{\beta(r)}(f;q;x)=\left\{\prod_{j=1}^{r}(u_n^{(j)}x;q)_n\right\}\cdot\\ \sum_{m=0}^{\infty}\sum_{k_1+\dotsb+k_r=m}\left\{f\left(\frac{1-q^{k_r}}{1-q^{n+k_r-1}}\right)\prod_{i=1}^{r}\frac{(u_n^{(j)})^{k_i}}{(q;q)_{k_i}}(q^n;q)_{k_i}\right\}x^m,\end{multlined} \tag{3}NEWLINE\]NEWLINE NEWLINEwhere \(f\in C[0,1]\), \(x\in[0,1]\) and \(n\in\mathbb{N}\). Observe that the operators in (3) are positive and linear for \(0<q<1\). First, the author gives a Krovkin type result for the operators (3) by means of \(A\)-statistical convergence. Then, the author computes the rates of \(A\)-statistical convergence of the operators with the help of the modulus of contunuity, the elements of the Lipschitz class, Pectre's \(K\)-functional and a Lipschitz-type maximal function, respectively.