ON CERTAIN SUMMATION-INTEGRAL TYPE OPERATORS

Abstract

Let S be the linear space of all real functions acting on a nonempty set X The operator M :S --->S is linear positive if: 1. M(a f + j3 g)= a M(f)+ M(g), where a and # are reals and f,gE S; 2. M(f)?_ 0 for any nonnegative function f E S. In 1885 [80], the German analyst Karl Weierstrass proved that any continuous real-valued function on a closed interval of the set of real numbers is the uniform limit on that interval of some sequence of polynomials with real coefficients. Since then, American analyst MIL Stone [80] has generalized the so-called "Weierstrass Approximation Theorem" to compact subsets of the set of real numbers, known as "Stone Weierstrass Theorem". For some further researches on this theorem we refer to (cf. eg. [19, 81, 98]). In 1912 [17], the Russian mathematician S.N. Bernstein gave another proof of Weierstrass approximation theorem by using a sequence of linear positive operators called the Bernstein polynomials. It is given as: 2 Chapter 0: Introduction and Contents of the Thesis If f is any bounded function on the interval [0,1], n -th order Bernstein polynomial is defined as: ( (11 xk (1— x)n-k x E [0,1] . 13,(f;x) = — k=0 n) \k/ The convergence of Bn(f;.) to f as n tends to infinity is slow but sure [105], allowing convergence of all derivatives and, of particular interest in design, preserving in a number of ways the shape of the graph of f . Moreover, an elegant asymptotic error estimate was found by E. Voronovskaja [101] in 1932.