In mathematics, the Riesz–Fischer theorem in real analysis is any of 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.
For many authors, the Riesz–Fischer theorem refers to the fact that the Lp spaces
The most common form of the theorem states that a measurable function on
is square integrable if and only if the corresponding Fourier series converges in the Lp space
This means that if the Nth partial sum of the Fourier series corresponding to a square-integrable function f is given by
is a two-sided sequence of complex numbers (that is, its indices range from negative infinity to positive infinity) such that
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–Fischer theorem (Dunford & Schwartz 1958, §IV.16).
Among them is the theorem that, if A is an orthonormal set in a Hilbert space H, and
Furthermore, if A is an orthonormal basis for H and x an arbitrary vector, the series
Moreover, the following conditions on the set A are equivalent: Another result, which also sometimes bears the name of Riesz and Fischer, is the theorem that
be an orthonormal system in R (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).
Combined with the Bessel's inequality, we know the converse as well: if f is a function in R, then the Fourier coefficients
In his Note, Riesz (1907, p. 616) states the following result (translated here to modern language at one point: the notation
Today, this result of Riesz is a special case of basic facts about series of orthogonal vectors in Hilbert spaces.
Riesz's Note appeared in March.
In May, Fischer (1907, p. 1023) states explicitly in a theorem (almost with modern words) that a Cauchy sequence in
Here is the statement, translated from French: Fischer goes on proving the preceding result of Riesz, as a consequence of the orthogonality of the system, and of the completeness of
It uses the fact that the indefinite integrals of the functions gn in the given Cauchy sequence, namely
for the Cauchy sequence is obtained by applying to G differentiation theorems from Lebesgue's theory.
Riesz uses a similar reasoning in his Note, but makes no explicit mention to the completeness of
He says that integrating term by term a trigonometric series with given square summable coefficients, he gets a series converging uniformly to a continuous function F with bounded variation.
For some authors, notably Royden,[1] the Riesz-Fischer Theorem is the result that
The proof below is based on the convergence theorems for the Lebesgue integral; the result can also be obtained for
the Minkowski inequality implies that the Lp space
the Minkowski inequality and the monotone convergence theorem imply that
requires some modifications, because the p-norm is no longer subadditive.
reduces to a simple question about uniform convergence outside a