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Fourier Series of Summable Functions

  • Adriaan C. Zaanen
Part of the Universitext book series (UTX)

Abstract

Given fL1(π,µ)the Fourier coefficients (c n : n = 0, ±1, ±2,…) of f were introduced in Definition 8.1 by defining
$${c_n} = {(2\pi )^{ - 1}}\int\limits_\Delta {f(x){e^{ - inx}}} dx,$$
(1)
where Δ is any interval of length 2π. To indicate that the Fourier coeffi­cients are those of the function f, the notation c n (f) does sometimes occur. Frequently the notation fˆ(n) instead of cn(f) is also used. The sequence (fˆ(n) : n = 0, ±1, ±2,…) is then denoted by fˆ. For any fL1(ℝ,µ) there is an analogous notion, although now it is not a sequence of numbers but again a function defined on the whole of ℝ. Precisely formulated, for fL1(ℝ,µ) the Fourier transform fˆ of f is the function, defined for any x ∈ ℝ by
$${{f}^{{\left( x \right)}}} = \int\limits_{\mathbb{R}} {f(y){{e}^{{ - ixy}}}} dy.$$
(2)

Keywords

Hilbert Space Fourier Series Fourier Coefficient Pointwise Convergence Summable Function 
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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