# Fourier Series of Continuous Functions

Part of the Universitext book series (UTX)

## Abstract

We begin the present section with some simple definitions (probably already known to most readers). For m, n integers the so-called Kronecker delta δ mn is defined by δ mn = 1 if m = n and δ mn = 0 if mn. For our second definition, let (f n : n = 0, ±1, ±2,...) be a set of real or complex functions, defined on the subset Δ of ℝ k . The set (f n ) is called an orthonormal set (or orthonormal system) on Δ if
$$f_n^ -$$
(1)
is the complex conjugate of f n and dx stands for dx1dx k Of course, the definition makes sense only if the integral of f m
$${f_m}f_n^ -$$
(3)
exists for all m, n. The definition is analogous if m and n are restricted to 0,1,2,… or to 1,2,… . If it is only given that
$$\int {_\Delta } fmf_n^ - dx = 0form \ne n,$$
(4)
then (fn) is said to be an orthogonal system on Δ. We immediately mention an example. For n = 0, ±1, ±2,…, let e n (x) = (2π)-1/2einx on ℝ. The system (e n : n = 0, ±1, ±2,…) is orthonormal on any interval [a, a + 2π], i.e., on any interval of length 2π in ℝ. The proof is immediate by observing that
$$2\pi \left\{ {{e_m}\left( x \right)\overline {{e_n}\left( x \right)} } \right\} = {e^{i\left( {m - n} \right)x + i\sin (m - n)x}}.$$
(5)

Sine