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Fourier series

  • M. H. Protter
  • C. B. MorreyJr.
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

In the study of power series in Chapter 9, we saw that an analytic function f can be represented by a power series
$$f(x) = \sum\limits_{n = 0}^\infty {{c_n}{{(x - a)}^n}}$$
(10.1)
for all values of x within the radius of convergence of the series. We recall that f has derivatives of all orders and that the coefficients c n in (10.1) are given by f (n) (a)/n!. In this chapter we shall be interested in series expansions of functions which may not be smooth. That is, we shall consider functions which may have only a finite number of derivatives at some points and which may be discontinuous at others. Of course, in such cases it is not possible to have expansions in powers of (x — a) such as (10.1). To obtain representations of unsmooth functions we turn to expansions in terms of trigonometric functions such as
$$1,{\kern 1pt} \cos x,\cos 2x,...,\cos nx,...$$
$${\kern 1pt} \sin x,\sin 2x,...,\sin nx,...$$

Keywords

Fourier Series Fourier Coefficient Piecewise Smooth Piecewise Continuous Function Sine Series 
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, New York Inc. 1977

Authors and Affiliations

  • M. H. Protter
    • 1
  • C. B. MorreyJr.
    • 1
  1. 1.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA

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