Representation Theorems

Abstract

Suppose we are given a trigonometric series

$$\sum\limits_{k = - \infty }^\infty {f(k)e^{ikx} }$$

((6.0.1))

with arbitrary complex coefficients f(k). How can one tell whether this series is the Fourier series of an X_2π,-function, in other words, whether the numbers f(k) are the Fourier coefficients gˆ(k) of some function g ϵ X_2π? The problem may be restated as follows: Given an arbitrary function f on ℤ, to determine conditions under which f admits a representation as the Fourier transform gˆ of some function g ∈X_2π, or as the finite Fourier-Stieltjes transform of some µ ∈ BV _2π . We have seen that f on ℤ has to satisfy certain necessary conditions in order to be the finite Fourier or Fourier- Stieltjes transform. Thus f on ℤ must be bounded in view of (4.1.2) and Prop. 4.3.2(ii). On the other hand, for L ²_2π we already know that a necessary and sufficient condition for a function f on ℤ to be the finite Fourier transform of some g ∈ L ²_2π , is that f ∈|². This is a consequence of the Parseval equation and the theorem of Riesz-Fischer. But nothing as simple seems possible for other classes such as L ^p_2π , p≠2.