Fourier series

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


In the study of power series in Chapter 9, we saw that an analytic function f can be represented by a power series
$$f(x) = \sum\limits_{n = 0}^\infty {{c_n}{{(x - a)}^n}}$$
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
$$1,{\kern 1pt} \cos x,\cos 2x,...,\cos nx,...$$
$${\kern 1pt} \sin x,\sin 2x,...,\sin nx,...$$


Fourier Series Fourier Coefficient Piecewise Smooth Piecewise Continuous Function Sine Series 
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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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