Rate of convergence on Baskakov-Beta-Bézier operators for bounded variation functions
Considered is the modification of classical Baskakov operators \[ V_n(f,x)=\sum\limits^{\infty}_{k=0} P_{n,k}(x)f\left(\frac{k}{n}\right), \] where \[ P_{n,k}(t)=\frac{n+k-1}{k}\frac{x^k}{(1+x)^{n+k}} \] for approximation Lebesgue integrable functions on \([0,\infty)\). Introduced is the Baskakov-Beta-Bézier operators \[ B_{n,\alpha}(f,x)=\sum\limits^{\infty}_{k=0} Q^{(\alpha)}_{n,k}(x) \int\limits^{\infty}_0 b_{n,k}(t)f(t)\,dt, \quad x \in [0,\infty), \] where \[ Q^{(\alpha)}_{n,k}(x)=J^{(\alpha)}_{n,k}(x)-J^{(\alpha)}_{n,k+1}(x), \] \[ J_{n,k}(x)=\sum\limits^{\infty}_{j=k} P_{n,j}(x), \] \[ b_{n,k}(t)=t^k/(B(k+1,n)(1+t)^{n+k+1}). \] The author received an estimate of the rate of convergence of \(B_{n,\alpha}(f,x)\) to functions of bounded variation. An open problem is formulated.
scientific article; zbMATH DE number 1852900