Integral Representations of Deviations of Linear Means of Fourier Series

Abstract

Assume that f(x) is a given function and we are going to approximate it by a sequence of functions F _n (f,x). Let δ _n (x)= δ _n (f,F _n ,x) =f(x)-F _n (f,x). If F _n (f,x) is a polynomial, then the value δ _n (x) can often be represented as a convolution where the function ϕ depends only on f, and the kernel Φ _n depends only on a method of constructing operators F _n (x). This forms a basis of one of the most widely used and efficient approaches to studying the value δ _n (x), which is finally reduced to investigating sequences of integral operators.