Riesz–Fischer theorem

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 L^p spaces from Lebesgue integration theory are complete.

Modern forms of the theorem

The most common form of the theorem states that a measurable function on [–π, π] is square integrable if and only if the corresponding Fourier series converges in the space L². This means that if the Nth partial sum of the Fourier series corresponding to a square-integrable function f is given by

S_N f(x) = \sum_{n=-N}^{N} F_n \, \mathrm{e}^{inx},

where F_n, the nth Fourier coefficient, is given by

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

then

\lim_{N \to \infty} \left \Vert S_N f - f \right \|_2 = 0,

where $\left \Vert \cdot \right \|_2$ is the L²-norm.

Conversely, if $\left \{ a_n \right \} \,$ 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.

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 x ∈ H, then

\langle x, y\rangle = 0

for all but countably many y ∈ A, and

\sum_{y\in A} |\langle x,y\rangle|^2 \le \|x\|^2.

Furthermore, if A is an orthonormal basis for H and x an arbitrary vector, the series

\sum_{y\in A} \langle x,y\rangle \, y

converges commutatively (or unconditionally) to x. This is equivalent to saying that for every ε > 0, there exists a finite set B₀ in A such that

\|x - \sum_{y\in B} \langle x,y\rangle y \| < \varepsilon

for every finite set B containing B₀. Moreover, the following conditions on the set A are equivalent:

the set A is an orthonormal basis of H
for every vector x ∈ H,

\|x\|^2 = \sum_{y\in A} |\langle x,y\rangle|^2.

Another result, which also sometimes bears the name of Riesz and Fischer, is the theorem that L² (or more generally L^p, 0 < p ≤ ∞) is complete.

Example

The Riesz–Fischer theorem also applies in a more general setting. Let R be an inner product space consisting of functions (for example, measurable functions on the line, analytic functions in the unit disc; in old literature, sometimes called Euclidean Space), and let { $\phi_n$ } 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). The theorem asserts that if the normed space R is complete (thus R is a Hilbert space), then any sequence { $c_n$ } that has finite ℓ² norm defines a function f in the space R.

The function f is defined by $f = \lim_{n \to \infty} \sum_{k=0}^n c_k \phi_k$ , limit in R-norm.

Combined with the Bessel's inequality, we know the converse as well: if f is a function in R, then the Fourier coefficients $(f,\phi_n)$ have finite ℓ² norm.

History: the Note of Riesz and the Note of Fischer (1907)

In his Note, Riesz (1907, p. 616) states the following result (translated here to modern language at one point: the notation L²([a, b]) was not used in 1907).

Let {φ_n} be an orthonormal system in L²([a, b]) and {a_n} a sequence of reals. The convergence of the series $\sum a_n^2$ is a necessary and sufficient condition for the existence of a function f such that

\int_a^b f(x) \varphi_n(x) \, \mathrm{d}x = a_n

for every n.

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 L²([a, b]) converges in L²-norm to some function f in L²([a, b]). In this Note, Cauchy sequences are called "sequences converging in the mean" and L²([a, b]) is denoted by Ω. Also, convergence to a limit in L²–norm is called "convergence in the mean towards a function". Here is the statement, translated from French:

Theorem. If a sequence of functions belonging to Ω converges in the mean, there exists in Ω a function f towards which the sequence converges in the mean.

Fischer goes on proving the preceding result of Riesz, as a consequence of the orthogonality of the system, and of the completeness of L².

Fischer's proof of completeness is somewhat indirect. It uses the fact that the indefinite integrals of the functions g_n in the given Cauchy sequence, namely

G_n(x) = \int_a^x g_n(t) \, \mathrm{d}t,

converge uniformly on [a, b] to some function G, continuous with bounded variation. The existence of the limit g ∈ L² 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 L², although his result may be interpreted this way. 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. The derivative f of F, defined almost everywhere, is square summable and has for Fourier coefficients the given coefficients.

Completeness of L^p, 0 < p ≤ ∞

The proof that L^p is complete is based on the convergence theorems for the Lebesgue integral.

When 1 ≤ p ≤ ∞, the Minkowski inequality implies that the space L^p is a normed space. In order to prove that L^p is complete, i.e. that L^p is a Banach space, it is enough (see e.g. Banach space#Definition) to prove that every series ∑ u_n of functions in L^p(μ) such that

\sum \|u_n\|_p < \infty

converges in the L^p-norm to some function f ∈ L^p(μ). For p < ∞, the Minkowski inequality and the monotone convergence theorem imply that

\int \Bigl( \sum_{n=0}^\infty |u_n| \Bigr)^p \, \mathrm{d}\mu \le \Bigl( \sum_{n=0}^{\infty} \|u_n\|_p \Bigr)^p< \infty, \ \ \text{ hence } \ \ f = \sum_{n=0}^\infty u_n

is defined μ–almost everywhere and f ∈ L^p(μ). The dominated convergence theorem is then used to prove that the partial sums of the series converge to f in the L^p-norm,

\int \left| f - \sum_{k=0}^{n} u_k \right|^p \, \mathrm{d}\mu \le \int \left( \sum_{\ell > n} |u_\ell| \right)^p \, \mathrm{d}\mu \rightarrow 0 \text{ as } n \rightarrow \infty.

The case 0 < p < 1 requires some modifications, due to the fact that the p-norm is no longer subadditive. One starts with the stronger assumption that

\sum \|u_n\|_p^p < \infty

and uses repeatedly that

\left|\sum_{k=0}^n u_k \right|^p \le \sum_{k=0}^n |u_k|^p \text{ when } p<1

The case p = ∞ reduces to a simple question about uniform convergence outside a μ-negligible set.

References