Fourier Series in L²

Abstract

It will be shown in this chapter that the problem of mean convergence of Fourier series in L² has a complete and simple solution. The abstract foundation for this situation lies in the fact that L² is a Hubert space with the inner (or scalar) product

$$\left( {f,g} \right) = {1 \over {2\pi }}\int {f\bar gdx,} $$

and that moreover the functions e_n defined by

$$e_n \left( x \right) = e^{inx} {\text{ }}\left( {n \in z} \right){\text{ }} $$

form an orthonormal base in L². This last means that the family (e_n) is orthonormal, in the sense that

$$\left( {e_m ,e_n } \right) = \delta _{{\text{mn}}} {\text{ }}\left( {m,n \in Z} \right), $$

and that

$$f \in {\text{L}}^2 ,\left( {f,e_n } \right) = 0\left( {n \in Z} \right) \Rightarrow f = 0{\text{ a}}.{\text{e}}. $$

Indeed, (8.3) is simply a restatement of the orthogonality relations, and the implication (8.4) is a special case of the uniqueness theorem 2.4.1. As Hubert space theory shows, these two facts imply that each f ∈ L² has a convergent expansion

$$f = \sum\limits_{n \in z} {\left( {f,{e_n}} \right){e_n};} $$

see, for example, [E], Corollary 1.12.5, or [HS], pp. 245-246, or [AB], pp. 239-240.