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The Fourier Transform

  • David F. Walnut
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

We have seen that if f(x) is a function supported on an interval [−L, L] for some L > 0, then f(x) can be represented by a Fourier series as
$$f\left( x \right) = \sum\limits_n {c\left( n \right){e^{2\pi in\left( {x/2L} \right)}}\quad where\quad c\left( n \right) = \frac{1}{{2L}}\int_{ - L}^L {f\left( t \right){e^{ - 2\pi it\left( {n/2L} \right)}}dt.} } $$
(3.1)

Keywords

Fourier Transform Fourier Series Prove Theorem Approximate Identity Fourier Inversion 
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 Science+Business Media New York 2004

Authors and Affiliations

  • David F. Walnut
    • 1
  1. 1.Department of Mathematical SciencesGeorge Mason UniversityFairfaxUSA

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