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Advanced Mathematics for Applications by Andrea Prosperetti

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9

Fourier Series: Theory

A pragmatic view of the Fourier series together with numerous examples was presented in Chapter 3. Here we give more details on the underlying theory which is important not only in itself, but also because the point-wise convergence properties of many other infinite eigenfunction series expansions can be reduced essentially to those of the Fourier series for the reasons mentioned at the end of §15.2 p. 364.

The Fourier series may also be viewed as just one instance of the expansion of the elements, or vectors, of a Hilbert space onto an orthogonal basis. For this reason we defer the treatment of several aspects related to the global convergence of the series to the general exposition of the theory of Hilbert spaces ...

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