Baskakov operators

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

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

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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