Riesz–Fischer theorem

In mathematics, the Riesz–Fischer theorem in real analysis states that a function is square integrable if and only if the corresponding Fourier series converges uniformly in the space ${\displaystyle L^{2}}$.

This means that if the Nth partial sum of the Fourier series corresponding to a function ${\displaystyle f}$ is given by

${\displaystyle S_{N}f(x)=\sum _{n=-N}^{N}F_{n}\,e^{inx}}$,

where ${\displaystyle F_{n}}$, the nth Fourier coefficient, is given by

${\displaystyle F_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)\,e^{-inx}\,dx}$,

then

${\displaystyle \lim _{n\to \infty }\left\Vert S_{n}f-f\right\|=0}$,

where ${\displaystyle \left\Vert g\right\|}$ is the ${\displaystyle L^{2}}$-norm, expressed as

${\displaystyle \left\Vert g\right\|=\int _{-2\pi }^{2\pi }g^{2}\,dx}$.

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

${\displaystyle \sum _{n=1}^{\infty }\left|a_{n}\right\vert ^{2}<\infty }$,

then there exists a function ${\displaystyle f}$ such that ${\displaystyle f}$ is square-integrable and the values ${\displaystyle a_{n}}$ are the Fourier coefficients of ${\displaystyle f}$.

The Riesz–Fischer theorem is a stronger form of Bessel's inequality, and can be used to prove Parseval's identity.

Hungarian mathematician Frigyes Riesz and Austrian mathematician Ernst Fischer, working independently, both discovered the theorem in 1907.

References

• Beals, Richard (2004). Analysis: An Introduction. New York: Cambridge University Press. ISBN 0-521-60047-2.
• Weisstein, Eric W. Reisz-Fischer Theorem. Retrieved May 3, 2005.