Riesz–Fischer theorem

In mathematics, the Riesz–Fischer theorem in real analysis refers to a number of closely related results concerning the properties of the space L² of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer.

The most common form of the theorem states that a function is square integrable if and only if the corresponding Fourier series converges in the space $l^2$. This means that if the "N"th partial sum of the Fourier series corresponding to a function $f$ is given by

:$S\_N\; f(x)\; =\; sum\_\{n=-N\}^\{N\}\; F\_n\; ,e^\{inx\},$

where $F\_n$, the "n"th Fourier coefficient, is given by

:$F\_n\; =frac\{1\}\{2pi\}int\_\{-pi\}^pi\; f(x),e^\{-inx\},dx$,

then

:$lim\_\{n\; o\; infty\}\; left\; Vert\; S\_n\; f\; -\; f\; ight\; |\_2\; =\; 0,$

where $left\; Vert\; cdot\; ight\; |\_2$ is the $L^2$-norm.

Conversely, if $left\; \{\; a\_n\; ight\; \}\; quad$ is a two-sided sequence of complex numbers (that is, its indices range from negative infinity to positive infinity) such that

:

then there exists a function $f$ such that $f$ is square-integrable and the values $a\_n$ are the Fourier coefficients of $f$.

This form of the Riesz–Fischer theorem is a stronger form of Bessel's inequality, and can be used to prove Parseval's identity for Fourier series.

Other results are often called the Riesz-Fisher theorem harv|Dunford|Schwartz|1958|loc=§IV.16. Among them is the theorem that, if "A" is an orthonormal set in a Hilbert space "H", and "x" ∈ "H", then:$langle\; x,\; y\; angle\; =\; 0$for all but countably many "y" ∈ "A", and the series:$sum\_\{yin\; A\}\; langle\; x,y\; angle\; y$
converges normally to "x". Moreover, the following conditions on the set "A" are equivalent:
* the set "A" is an orthonormal basis of "H"
* $|x|^2\; =\; sum\_\{yin\; A\}\; |langle\; x,y\; angle|^2.$

Another result, which also sometimes bears the name of Riesz and Fisher, is the theorem that L² is complete.

Example

The Riesz-Fischer theorem also applies in a more general setting. Let $R$ be an inner product space (in old literature, sometimes called Euclidean Space), and let {$phi\_n$} be an orthonormal system (e.g. Fourier basis, Hermite or Laguerre polynomials, etc. -- see orthogonal polynomials), not necessarily complete (in an inner product space, an orthonormal set is complete if no nonzero vector is orthogonal to every vector in the set). The theorem asserts that if $R$ is complete, then any sequence {$c\_n$} that has finite $l^2$ norm defines a function $f$ in $L^2$, e.g. $f$ is square-integrable.

The function $f$ is defined$f\; =\; lim\_\{n\; o\; infty\}\; sum\_\{k=0\}^n\; c\_k\; phi\_k$.

Combined with the Bessel's inequality, we know the converse as well: if $f$ is square-integrable, then the Fourier coefficients $(f,phi\_n)$ have finite $l^2$ norm.

References

*citation|last=Beals|first=Richard|year=2004|title=Analysis: An Introduction|publication-place=New York|publisher=Cambridge University Press|isbn=0-521-60047-2.
*.
*citation|last=Fischer|first=Ernst|authorlink=Ernst Sigismund Fischer|title=Sur la convergence en moyenne|journal=Comptes rendus de l'Académie des sciences|volume=144|pages=1148-1151|year=1907.
*citation|last=Riesz|first=Frigyes|authorlink=Frigyes Riesz|title=Sur les systèmes orthogonaux de fonctions|journal=Comptes rendus de l'Académie des sciences|year=1907|volume=144|pages=615-619.