## Abstract

In this chapter we tell some basic facts about functionals and operators in spaces \(L_p\) and apply them to Fourier series in \(L_p[0, 2\pi ]\) and Fourier transform in \(L_p(-\infty , \infty )\). The contents of the chapter: the Hölder inequality; connections between the spaces \(L_p\) for different values of *p*; weighted integration functionals; the general form of linear functionals on \(L_p\); \(\delta \)-sequences and the Dini theorem; the Fourier transform in \(L_1(-\infty , \infty )\); inversion formulas; the Fourier transform and differentiation; the Fourier transform in \(L_2(-\infty , \infty )\); the Hadamard three-lines theorem; the Riesz–Thorin theorem; applications to Fourier series and the Fourier transform.

## Copyright information

© Springer International Publishing AG, part of Springer Nature 2018