Riesz–Fischer theorem

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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 from Lebesgue integration theory are complete.

Modern forms of the theorem

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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   where   the nth Fourier coefficient, is given by   then   where   is the  -norm.

Conversely, if   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   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   then   for all but countably many   and   Furthermore, if A is an orthonormal basis for H and x an arbitrary vector, the series   converges commutatively (or unconditionally) to x. This is equivalent to saying that for every   there exists a finite set   in A such that   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  
 

Another result, which also sometimes bears the name of Riesz and Fischer, is the theorem that   (or more generally  ) is complete.

Example

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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   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   that has finite   norm defines a function f in the space R.

The function f is defined by   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   have finite   norm.

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

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

Let   be an orthonormal system in   and   a sequence of reals. The convergence of the series   is a necessary and sufficient condition for the existence of a function f such that  

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   converges in  -norm to some function   In this Note, Cauchy sequences are called "sequences converging in the mean" and   is denoted by   Also, convergence to a limit in  –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   converges in the mean, there exists in   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  

Fischer's proof of completeness is somewhat indirect. It uses the fact that the indefinite integrals of the functions gn in the given Cauchy sequence, namely   converge uniformly on   to some function G, continuous with bounded variation. The existence of the limit   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   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 Lp,  0 < p ≤ ∞

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For some authors, notably Royden,[1] the Riesz-Fischer Theorem is the result that   is complete: that every Cauchy sequence of functions in   converges to a function in   under the metric induced by the p-norm. The proof below is based on the convergence theorems for the Lebesgue integral; the result can also be obtained for   by showing that every Cauchy sequence has a rapidly converging Cauchy sub-sequence, that every Cauchy sequence with a convergent sub-sequence converges, and that every rapidly Cauchy sequence in   converges in  

When   the Minkowski inequality implies that the Lp space   is a normed space. In order to prove that   is complete, i.e. that   is a Banach space, it is enough (see e.g. Banach space#Definition) to prove that every series   of functions in   such that   converges in the  -norm to some function   For   the Minkowski inequality and the monotone convergence theorem imply that   is defined  –almost everywhere and   The dominated convergence theorem is then used to prove that the partial sums of the series converge to f in the  -norm,  

The case   requires some modifications, because the p-norm is no longer subadditive. One starts with the stronger assumption that   and uses repeatedly that   The case   reduces to a simple question about uniform convergence outside a  -negligible set.

See also

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References

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  1. ^ Royden, H. L. (13 February 2017). Real analysis. Fitzpatrick, Patrick, 1946- (Fourth ed.). New York, New York. ISBN 9780134689494. OCLC 964502015.{{cite book}}: CS1 maint: location missing publisher (link)
  • Beals, Richard (2004), Analysis: An Introduction, New York: Cambridge University Press, ISBN 0-521-60047-2.
  • Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.
  • Fischer, Ernst (1907), "Sur la convergence en moyenne", Comptes rendus de l'Académie des sciences, 144: 1022–1024.
  • Riesz, Frigyes (1907), "Sur les systèmes orthogonaux de fonctions", Comptes rendus de l'Académie des sciences, 144: 615–619.