Riesz–Fischer theorem

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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 in the space $l 2$ .

This means that if the Nth 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 nth Fourier coefficient, is given by

$F_n =\frac{1}{2\pi}\int_{-\pi}^\pi f(x)\,e^{-inx}\,dx$ ,

then

$\lim_{n \to \infty} \left \Vert S_n f - f \right \|_2 = 0,$

where $\left \Vert \cdot \right \|_2$ is the $L 2$ -norm.

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

$\sum_{n=-\infty}^\infty \left | a_n \right \vert^2 < \infty,$

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

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

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

[edit] Generalization

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 { $φ n$ } 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 system is complete if and only if it is closed). 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 $f = \lim_{n \to \infty} \sum_{k=0}^n c_k \phi_k$ .

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