Riesz-Fischer theorem

Riesz-Fischer theorem

[′rēsh ′fish·ər ‚thir·əm]
(mathematics)
The vector space of all real- or complex-valued functions whose absolute value squared has a finite integral constitutes a complete inner product space.
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By the Riesz-Fischer theorem, we can extract from {[v.sub.m]} a subsequence still denoted by {[v.sub.m]}, such that
which by the generalized Riesz-Fischer theorem imply the existence of functions [mathematical expression not reproducible] and [mathematical expression not reproducible] whose Fourier coefficients are {[[gamma].sub.nk] : n [member of] N} and {[[gamma].sub.n] : n [member of] [N.sub.0]}, respectively.
This theorem entails the use of the Riesz-Fischer theorem, the Planeherel-Polya inequality, and a uniqueness theorem which in turn depends on properties of the indicator diagram for entire functions.
To this end, since [summation over (k[greater than or equal to]1]1/[k.sup.2][(ln (k + 1))).sup.4] < [infinity], then by the Riesz-Fischer theorem [5, p193], the series