7Fourier Series: Basic Concepts

DOI: 10.1201/9781003214526-7

  • The idea of Fourier series

  • Calculating a Fourier series

  • Convergence of Fourier series

  • Odd and even functions

  • Fourier series on arbitrary intervals

  • Orthogonality

7.1 Fourier Coefficients

Trigonometric and Fourier series constitute one of the oldest parts of analysis. They arose, for instance, in classical studies of the heat and wave equations. Today they play a central role in the study of sound, heat conduction, electromagnetic waves, mechanical vibrations, signal processing, and image analysis and compression. Whereas power series (see Chapter 4) can only be used to represent very special functions (most functions, even smooth ones, do not have convergent power series), Fourier ...

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