Abstract
It will be shown in this chapter that the problem of mean convergence of Fourier series in L2 has a complete and simple solution. The abstract foundation for this situation lies in the fact that L2 is a Hubert space with the inner (or scalar) product
and that moreover the functions en defined by
form an orthonormal base in L2. This last means that the family (en) is orthonormal, in the sense that
and that
Indeed, (8.3) is simply a restatement of the orthogonality relations, and the implication (8.4) is a special case of the uniqueness theorem 2.4.1. As Hubert space theory shows, these two facts imply that each f ∈ L2 has a convergent expansion
see, for example, [E], Corollary 1.12.5, or [HS], pp. 245-246, or [AB], pp. 239-240.
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© 1979 Springer-Verlag New York, Inc.
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Edwards, R.E. (1979). Fourier Series in L2. In: Fourier Series. Graduate Texts in Mathematics, vol 64. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-6208-4_8
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DOI: https://doi.org/10.1007/978-1-4612-6208-4_8
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