Voronovskaya type asymptotic approximation by modified Gamma operators
Abstract In this paper we deal with the following modified Gamma operators A n ( f ; x ) = ∫ 0 ∞ K n ( x , t ) f ( t ) dt , defined for any f for which the last integral is convergent, where K n ( x , t ) = ( 2 n + 3 ) ! n ! ( n + 2 ) ! · t n · x n + 3 ( x + t ) 2 n + 4 , x , t ∈ ( 0 , ∞ ) . We study some direct results in asymptotic approximation.
Modified gamma operators in Lp spaces
In this paper, we consider certain positive linear operators connected with the classical gamma operators in Lp (1 ≤ p ≤ ∞) spaces. We study approximation properties of these operators, including theorems on the degree of approximation. The results are given in terms of some Ditzian–Totik modulus of smoothness.
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.