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V.A. Baskakov [a2] introduced a sequence of linear positive operators $L _ {n}$ with weights

$$p _ {nk } ( x ) = ( - 1 ) ^ {k} { \frac{x ^ {k} }{k! } } \phi _ {n} ^ {( k ) } ( x )$$

by

$$\tag{a1 } ( L _ {n} f ) ( x ) = \sum _ {k = 0 } ^ \infty p _ {nk } ( x ) f \left ( { \frac{k}{n} } \right ) ,$$

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

i) $\phi _ {n} \in C ^ \infty [ 0,b ]$;

ii) $\phi _ {n} ( 0 ) = 1$;

iii) $\phi _ {n}$ is completely monotone, i.e., $( - 1 ) ^ {k} \phi _ {n} ^ {( k ) } \geq 0$;

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

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,

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

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

where $f$ is a function on $[ 0, \infty )$ 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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