Rate of convergence of bounded variation functions by a Bézier-Durrmeyer variant of the Baskakov operators (Q1774675)

Let \(W(0,\infty)\) be the class of functions \(f\) which are locally integrable on \((0,\infty)\) and are of polynomial growth as \(t\to\infty\), that is, for some positive \(r\), there holds \(f(t)= O(t^r)\), \(t\to\infty\). The Durrmeyer variant \(\widetilde V_n\) of the well-known Baskakov operators associates to each function \(f\in W(0,\infty)\) the series \[ \widetilde V_n(f,x)= (n-1) \sum^\infty_{k=0} p_{nkk}(x) \int^\infty_0 p_{n,k}(t) f(t)\,dt, \] where \(x\in [0,\infty)\) and \(p_{n,k}(x)= {n+k-1\choose k}x^k(1-x)^{-n-k}\). For each function \(f\in W(0,\infty)\) and \(\alpha\geq 1\), the Bézier-type Baskakov-Durrmeyer operators \(\widetilde V_{n,\alpha}\) are defined as follows: \[ \widetilde V_{n,\alpha}(f, x)= (n-1) \sum^\infty_{k=0} Q^{(\alpha)}_{n,k}(x) \int^\infty_0 p_{n,k}(t) f(t)\,dt, \] where \(Q^{(\alpha)}_{n,k}(x)= J^\alpha_{nkk}(x)- J^\alpha_{n,k+1}(x)\) and \(J_{n,k}(x)= \sum^\infty_{j=k} p_{n,j}(x)\). The main result is the following: Theorem. Assume that \(f\in W(0,\infty)\) is a function of bounded variation on every finite subinterval of \((0,\infty)\). Furthermore, let \(\alpha\geq 1\), \(\lambda> 2\) and \(x\in (0,\infty)\) be given. Then, for each \(r\in\mathbb{N}\), there exists a constant \(M(f,\alpha,r,x)\) such that for sufficiently large \(n\), the Bézier-type Baskakov-Durrmeyer operators \(\widetilde V_{n,\alpha}\) satisfy the estimate \[ \begin{multlined}\Biggl|\widetilde V_{n,\alpha}(f,x)- \Biggl[{1\over \alpha+1} f(x+)+ {\alpha\over \alpha+1} f(x-)\Biggr]\Biggr|\leq\\ {\alpha(10+11x)\over 2\sqrt{n\times (1+x)}} |f(x+)- f(x-)|+ {2\alpha\lambda(1+ x)+x\over nx} \sum^n_{k=1} \bigvee^{x+ x/\sqrt{k}}_{x-x/\sqrt{k}}(g_x)+ {M(f,\alpha, r,x)\over n^r},\end{multlined} \] where \[ q_x(t)= \begin{cases} f(t)- f(x-),\quad &\text{if }0\leq t< x\\ 0,\quad &\text{if }t= x\\ f(t)- f(x+),\quad &\text{if }x< t<\infty\end{cases} \] and \(\bigvee^b_a (g_x)\) is the total variation of \(g_x\) on \([a,b]\).