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On the Bézier variant of Srivastava-Gupta operators. (English) Zbl 1084.41018

Let \(B_r[0,\infty)\) be the class of bounded variation functions satisfying the growth condition \(|f(t)|\leq M(1+t)^r\), \(M>0\), \(r\geq 0\), \(t\to\infty\). For a function \(f\in B_r[0,\infty)\), the operators \(G_{n,c}\) are defined by \[ G_{n,c}(f,x)=n\sum^\infty_{k=1} p_{n,k}(x,c)\int^\infty_0 p_{n+c,k-1}(t,c)f(t)\,dt+p_{n,0}(x,c)f(0), \] where \(p_{n,k}(x,c)=\frac{(-x)^k}{k!} \phi^{(k)}_{n,c}(x)\) and \(\phi^{(k)}_{n,c}(x)=\begin{cases} e^{-nx}, & c=0\\(1+cx)^{-n/c},& c=1,2,3,\dots\end{cases}\) This general sequence of linear positive operators was introduces by H. M. Srivastava and V. Gupta. The authors have defined the Bézier variant of the operators \(G_{n,c}\) as follows: for \(f\in B_r[0,\infty)\) and for each \(\alpha\geq 1\) let \[ G_{n,c,\alpha}(f,x)=n\sum^\infty_{k=1}Q^{(\alpha)}_{n,k}(x,c)\int^\infty_0 p_{n+c,k-1}(t,c)f(t)\,dt+Q^{(\alpha)}_{n,0}(x,c)f(0), \] where \(Q^{(\alpha)}_{n,k}(x,c)={\mathcal F}^\alpha_{n,k}(x,c)-{\mathcal F}^\alpha_{n,k+1}(x,c)\) and \({\mathcal F}_{n,k}(x,c)=\sum^\infty_{j=k}p_{n,j}(x,c)\). The main result is an estimate of the rate of convergence of these operators for functions of bounded variation. Theorem. Let \(f\) be an element of \(B_r[0,\infty)\). If \(\alpha>1\), \(r\in \mathbb{N}\) and \(\lambda>2\) are given, then there exists a constant \(c(f,\alpha,r;x)\) such that for \(n\) sufficiently large \[ \begin{split} |G_{n,c,\alpha}(f,x)-\left[\frac{1}{\alpha+1}\;f(x+)+\frac{\alpha}{\alpha+1}\;f(x-)\right]\mid \leq 2\alpha\sqrt{\frac{1+cx}{2enx}}|f(x+)-f(x-)|+\frac{6\alpha\lambda x(1+cx)}{nx}\\ \sum^n_{k=1}V^{x+x/\sqrt k}_{x-x/\sqrt k}(g_x)+c(f,\alpha,r;x)\alpha 2^r\frac{(1+x)^r}{x^r}\;O(n^{-r}),\end{split} \] where \[ g_x(t)=\begin{cases} f(t)=f(x-),\quad &\text{if }0\leq t<x\\ 0, & \text{if }t=x\\ f(t)-f(x+),& \text{if }t>x\end{cases}, \] \(V^b_a(g_x)\) is the total variation of \(g_x\) on \([a,b]\) and \(O(n^{-r})=\mu_{n,2r}(x,c)\) is the moment of order \(2r\), connected with the operator \(G_{n,c}\).

MSC:

41A36 Approximation by positive operators
41A25 Rate of convergence, degree of approximation
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