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The Fourier Transforms

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Lecture Notes on Wavelet Transforms

Part of the book series: Compact Textbooks in Mathematics ((CTM))

Abstract

Historically, Joseph Fourier (1770–1830) first introduced the remarkable idea of expansion of a function in terms of trigonometric series without giving any attention to rigorous mathematical analysis. The integral formulas for the coefficients of the Fourier expansion were already known to Leonhard Euler (1707–1783) and others. In fact, Fourier developed his new idea for finding the solution of heat (or Fourier) equation in terms of Fourier series so that the Fourier series can be used as a practical tool for determining the Fourier series solution of partial differential equations under prescribed boundary conditions. Thus, the Fourier series of a function f(t) defined on the interval (−L, L) is given by

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Notes

  1. 1.

    The following list includes books and research papers that have been useful for the preparation of these notes as well as some which may be of interest for further study.

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, University of Texas – Rio Grande Valley, Edinburg, Texas, USA

    Lokenath Debnath

  2. Department of Mathematics, University of Kashmir, Anantnag, Jammu and Kashmir, India

    Firdous A. Shah

Authors
  1. Lokenath Debnath
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  2. Firdous A. Shah
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Debnath, L., Shah, F.A. (2017). The Fourier Transforms. In: Lecture Notes on Wavelet Transforms. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-59433-0_1

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