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].

References

[a1]	P.N. Agrawal, H.S. Kasana, "On simultaneous approximation by Szász–Mirakian operators" Bull. Inst. Math. Acad. Sinica , 22 (1994) pp. 181–188
[a2]	V.A. Baskakov, "An example of a sequence of linear positive operators in the space of continuous functions" Dokl. Akad. Nauk SSSR , 113 (1957) pp. 249–251 (In Russian)
[a3]	H. Berens, "Pointwise saturation of positive operators" J. Approx. Th. , 6 (1972) pp. 135–146
[a4]	M. Heilmann, "Approximation auf $[0,\infty)$ durch das Verfahren der Operatoren vom Baskakov–Durrmeyer Typ" , Univ. Dortmund (1987) (Dissertation)
[a5]	M. Heilmann, M.W. Müller, "On simultaneous approximation by the method of Baskakov–Durrmeyer operators" Numer. Funct. Anal. Optim. , 10 (1989) pp. 127–138
[a6]	C.P. May, "Saturation and inverse theorems for combinations of a class of exponential-type operators" Canad. J. Math. , 28 (1976) pp. 1224–1250
[a7]	S.P. Singh, "On Baskakov-type operators" Comment. Math. Univ. St. Pauli, , 31 (1982) pp. 137–142
[a8]	Y. Suzuki, "Saturation of local approximation by linear positive operators of Bernstein type" Tôhoku Math. J. , 19 (1967) pp. 429–453