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Fourier Series of Continuous Functions

  • Adriaan C. Zaanen
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)

Keywords

Fourier Series Fourier Coefficient Trigonometric Polynomial Trigonometric Series Finite Variation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Adriaan C. Zaanen
    • 1
  1. 1.Department of MathematicsUniversity of LeidenLeidenThe Netherlands

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