Some approximation properties of Baskakov-Durrmeyer-Stancu operators (Q422908)

The authors introduce a new type of Baskakov-Durrmeyer--Stancu (BDS) operators \(D_n^{(\alpha,\beta)}(f,x)\) defined by NEWLINE\[NEWLINED_n^{(\alpha,\beta)}(f,x)=\sum\limits^{\infty}_{k=1}p_{n,k}(x)\int\limits_0^{\infty}b_{n,k}(t) f\left(\frac{nt+\alpha}{n+\beta}\right)dt+p_{n,0}(x) f\left(\frac{\alpha}{n+\beta}\right), \tag{1} NEWLINE\]NEWLINE where \(\sum\limits^{\infty}_{k=0}p_{n,k}(x)=1, ~\int\limits_0^{\infty}p_{n,k}(x) dx=\frac{1}{n-1},\) \(\sum\limits^{\infty}_{k=0}b_{n,k}(x)=n+1, ~\int\limits_0^{\infty}b_{n,k}(x)=1.\) In Section 2 they estimate moments of these operators and also obtain the recurrence relations for the moments. Some approximation properties and asymptotic formulae for the operators (1) are received in Section 3. In Section 4 better error estimations for operators (1) using the King type approach are given.