Function Vector Spaces and Fourier Series
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 2π. 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.
KeywordsFourier Series Function Vector Fourier Coefficient Dirichlet Condition Integral Kernel
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