Baskakov operators

V.A. Baskakov [a2] introduced a sequence of linear positive operators $L_n$ with weights

$$p_{nk}(x)=(-1{)}^k\frac{x^k}{k!}{\varphi }_n^{(k)}(x)$$

by

$$\begin{array}{}\text{(a1)}& (L_{n}f)(x)=\sum _{k=0}^{\mathrm{\infty }}{p}_{nk}(x)f\left(\frac{k}{n}\right),\end{array}$$

where $n\in \mathbf{N}$, $x\in [0,b]$, $b>0$, for all functions $f$ on $[0,\mathrm{\infty })$ for which the series converges. Here, $\{{\varphi }_n{\}}_{n\in \mathbf{N}}$ is a sequence of functions defined on $[0,b]$ having the following properties for every $n,k\in \mathbf{N}$, $k>0$:

i) ${\varphi }_n\in C^{\mathrm{\infty }}[0,b]$;

ii) ${\varphi }_n(0)=1$;

iii) ${\varphi }_n$ is completely monotone, i.e., $(-1{)}^k{\varphi }_n^{(k)}\ge 0$;

iv) there exists an integer $c$ such that ${\varphi }_n^{(k+1)}=-n{\varphi }_{n+c}^{(k)}$, $n>max\{0,-c\}$.

Baskakov studied convergence theorems of bounded continuous functions for the operators (a1). For saturation classes for continuous functions with compact support, see [a8]. For a result concerning bounded continuous functions, see [a3].

In his work on Baskakov operators, C.P. May [a6] took conditions slightly different from those mentioned above and showed that the local inverse and saturation theorems hold for functions with growth less than $(1+t{)}^N$ for some $N>0$. Bernstein polynomials and Szász–Mirakian operators are the particular cases of Baskakov operators considered by May.

S.P. Singh [a7] studied simultaneous approximation, using another modification of the conditions in the original definition of Baskakov operators. However, it was shown that his result is not correct (cf., e.g., [a1], Remarks).

Motivated by the Durrmeyer integral modification of the Bernstein polynomials, M. Heilmann [a4] modified the Baskakov operators in a similar manner by replacing the discrete values $f(k/n)$ in (a1) by an integral over the weighted function, namely,

$$(M_{n}f)(x)=\sum _{k=0}^{\mathrm{\infty }}{p}_{nk}(x)(n-c)\underset{0}{\overset{\mathrm{\infty }}{\int }}{p}_{nk}(t)f(t)dt,$$

$$n>c,x\in [0,\mathrm{\infty }),$$

where $f$ is a function on $[0,\mathrm{\infty })$ for which the right-hand side is defined. He studied global direct and inverse $L_p$- approximation theorems for these operators.

Subsequently, a global direct result for simultaneous approximation in the $L_p$- metric in terms of the second-order Ditzian–Totik modulus of smoothness was proved, see [a5]. For local direct results for simultaneous approximation of functions with polynomial growth, see [a5].

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