Fourier Series of Summable Functions

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


Given fL1(π,µ)the Fourier coefficients (c n : n = 0, ±1, ±2,…) of f were introduced in Definition 8.1 by defining
$${c_n} = {(2\pi )^{ - 1}}\int\limits_\Delta {f(x){e^{ - inx}}} dx,$$
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
$${{f}^{{\left( x \right)}}} = \int\limits_{\mathbb{R}} {f(y){{e}^{{ - ixy}}}} dy.$$


Hilbert Space Fourier Series Fourier Coefficient Pointwise Convergence Summable Function 
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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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