The Θ‐transformation of certain positive linear operators

Abstract

ABSTRACT. The intention of this paper is to describe a const,'uction method for a new sequence of linear positive operators, which enables us to get a pointwise order of approximation regarding the polynomial sun'tmator operators which have "best" properties of approximation.KEY WORDS AND PHRASES. Approximation by positive linear operators, discrete linear operators, (C, 1) means of Chebyshev series.1. The aim of this paper can be described in the following way" Starting wih a sequence A (A,) of approximation operators, we construct by means of the so called 0 transformation a new sequence of operators B (B=) O(A). With the known properties of A we get the corresponding properties of the sequence B O(A). We also prove, that if A is the sequence of (C, 1) means of Chebyshev series, the polynomials (Bf), f C(I), furnish a pointwise order of approximation similar to the best order of approximation.Let H,, n 0, be the linear space of all algebraic polynomials with real coefficients of degree 5 n and T,(t) cos(, arccos t) the n th Chebyshev polynomial, n 0. Further for f X and a polynomial 9 we use the inner product <f, 9) / f(t)9(t)w(t)dt.