Function Vector Spaces and Fourier Series

  • Kurt Bernardo Wolf
Part of the Mathematical Concepts and Methods in Science and Engineering book series (MCSENG, volume 11)


Vector spaces of functions can be infinite-dimensional. This implies a nontrivial extension of many of the concepts developed for finite-dimensional spaces. Section 4.1 is meant to provide a general picture of the location and depth of these extensions, introducing an infinite orthonormal set of functions (2π)-1/2 exp(inx), for n = 0, ± 1, ± 2, …, periodic in x with period . A large class of functions can be expanded in a series, called Fourier series, involving this orthonormal set. In Section 4.2 we prove one version of the Dirichlet conditions which give a sufficiency definition for this set, while in Sections 4.3 and 4.4 we explore several properties of series expansions related to each other by translation, inversion, complex conjugation, and differentiation and examine their convergence rates and the Gibbs phenomenon. The next two sections, 4.5 and 4.6, enter into the field of generalized functions and their divergent series representation. Although the complete mathematical treatment of this subject is by no means elementary, we have followed a “middle path” in the spirit of a physicist's use of quantum mechanics. Section 4.7 collects some results to be used in Chapter 5 and establishes a link with Part III.


Fourier Series Function Vector Fourier Coefficient Dirichlet Condition Integral Kernel 
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Copyright information

© Springer Science+Business Media New York 1979

Authors and Affiliations

  • Kurt Bernardo Wolf
    • 1
  1. 1.Instituto de Investigaciones en Matemáticas Aplicadas y en SistemasUniversidad Nacional Autónoma de MéxicoMéxico D.F.México

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