On non-uniqueness of the order of saturation

Abstract

1. Let X be a normed linear space of periodic functions (of period 2π), which includes the class of trigonometric polynomials. We restrict ourselves throughout to functions f ∈ X. Let D) = (dn, k) be the matrix of a regular sequence-to-sequence transformation. (We could also consider sequence-to-function transformations, in which case n is replaced by a continuous variable.) We suppose that, for all n,Let {Ln(x)} be the D transform of the Fourier series of f(x). If f(x) is a constant, it follows from (1) that, for all n, Ln(x) = f(x). However, it is often found that, roughly speaking, except in this trivial case, Ln(x) cannot tend to f(x) (in the topology given by the norm) with more than a certain degree of rapidity. This leads to the concept of ‘saturation’, which was first introduced by Favard (1) and Zamanski (8) and which has since been investigated by many authors (see (2), (3), (4), (5), (6), (7)). This is defined as follows.