Introducción a las series de Fourier no armónicas

Abstract

In the present document is desired to generalize the traditional Harmonic Fourier Series over a Hilbert space. First some concepts and results of functional analysis that will be useful are established. Then the concept of Frame is introduced, which is a tool in an advanced study of Non-Harmonic Fourier series. It follows with the study of the completeness of sequences in Hilbert spaces then in particular of the trigonometric system. Finally a reconstruction of the theory of Riesz Basis and Stability is done from the text “An Introduction to Non-Harmonic Fourier Series” of Robert M. Young, where Riesz basis are characterized and the Paley-Wiener theorem is introduced, which provides conditions so that the trigonometric system be stable in a Hilbert space under “sufficiently small” perturbations of integers.