Riesz–Fischer theorem
Encyclopedia
In mathematics
, the Riesz–Fischer theorem in real analysis
refers to 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.
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
where Fn, the nth Fourier coefficient
, is given by
then
where
is the L2-norm
.
Conversely, if
is a two-sided sequence
of complex number
s (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 . Among them is the theorem that, if A is an orthonormal set in a Hilbert space
H, and x ∈ H, then
for all but countably many y ∈ A, 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 ε > 0, there exists a finite set B0 in A such that
for every finite set B containing B0. 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 L2 (or more generally Lp, 0 < p ≤ ∞) is complete.
} 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 ℓ2 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 ℓ2 norm
.
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, 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:
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 gn in the given Cauchy sequence, namely
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.
.
When 1 ≤ p ≤ ∞, the Minkowski inequality
implies that the space Lp
is a normed space. In order to prove that Lp is complete, i.e. that Lp is a Banach space
, it is enough to prove that every series ∑ un of functions in Lp(μ) such that
converges in the Lp-norm to some function f ∈ Lp(μ). For p < ∞, the Minkowski inequality and the monotone convergence theorem imply that
is defined μ–almost everywhere and f ∈ Lp(μ). The dominated convergence theorem
is then used to prove that the partial sums of the series converge to f in the Lp-norm,
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
and uses repeatedly that 
when p < 1.
The case p = ∞ reduces to a simple question about uniform convergence outside a μ–negligible set.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Riesz–Fischer theorem in real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
refers to a number of closely related results concerning the properties of the space L2
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz
Frigyes Riesz
Frigyes Riesz was a mathematician who was born in Győr, Hungary and died in Budapest, Hungary. He was rector and professor at University of Szeged...
and Ernst Sigismund Fischer
Ernst Sigismund Fischer
Ernst Sigismund Fischer was a mathematician born in Vienna, Austria. He worked alongside both Mertens and Minkowski at the Universities of Vienna and Zurich, respectively...
.
For many authors, the Riesz–Fischer theorem refers to the fact that the Lp spaces
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
from Lebesgue integration
Lebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
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 ifIf and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
the corresponding Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
converges in the space L2
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
. This means that if the Nth partial sum of the Fourier series corresponding to a square-integrable function f is given by
where Fn, the nth Fourier coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
, is given by
then
where
is the L2-normNorm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
.
Conversely, if
is a two-sided sequenceSequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s (that is, its indices
Index (mathematics)
The word index is used in variety of senses in mathematics.- General :* In perhaps the most frequent sense, an index is a number or other symbol that indicates the location of a variable in a list or array of numbers or other mathematical objects. This type of index is usually written as a...
range from negative infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of 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
Bessel's inequality
In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence....
, and can be used to prove Parseval's identity
Parseval's identity
In mathematical analysis, Parseval's identity is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is thePythagorean theorem for inner-product spaces....
for Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
.
Other results are often called the Riesz-Fischer theorem . Among them is the theorem that, if A is an orthonormal set in a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
H, and x ∈ H, then
for all but countably many y ∈ A, 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 ε > 0, there exists a finite set B0 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 x ∈ H,
Another result, which also sometimes bears the name of Riesz and Fischer, is the theorem that L2 (or more generally Lp, 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 {} be an orthonormal system in R (e.g. Fourier basis, Hermite or Laguerre polynomialsLaguerre polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre ,are the canonical solutions of Laguerre's equation:x\,y + \,y' + n\,y = 0\,which is a second-order linear differential equation....
, etc. -- see orthogonal polynomials
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
), not necessarily complete (in an inner product space, an orthonormal set
Orthonormality
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and both of unit length. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length...
is complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
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
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
), then any sequence {
} that has finite ℓ2 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
Bessel's inequality
In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence....
, we know the converse as well: if f is a function in R, then the Fourier coefficients
have finite ℓ2 normNorm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
.
History: the Note of Riesz and the Note of Fischer (1907)
In his Note, states the following result (translated here to modern language at one point: the notation L2([a, b]) was not used in 1907).- Let {φn } be an orthonormal system in L2([a, b]) and {an } a sequence of reals. The convergence of the series
is a necessary and sufficient condition for the existence of a function f such that
- 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, states explicitly in a theorem (almost with modern words) that a Cauchy sequence
Cauchy sequence
In mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...
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 gn in the given Cauchy sequence, namely
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 Lp, 0 < p ≤ ∞
The proof that Lp is complete is based on the convergence theorems for the Lebesgue integralLebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
.
When 1 ≤ p ≤ ∞, the Minkowski inequality
Minkowski inequality
In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let S be a measure space, let 1 ≤ p ≤ ∞ and let f and g be elements of Lp...
implies that the space Lp
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
is a normed space. In order to prove that Lp is complete, i.e. that Lp is a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
, it is enough to prove that every series ∑ un of functions in Lp(μ) such that
converges in the Lp-norm to some function f ∈ Lp(μ). For p < ∞, the Minkowski inequality and the monotone convergence theorem imply that
is defined μ–almost everywhere and f ∈ Lp(μ). The dominated convergence theorem
Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions...
is then used to prove that the partial sums of the series converge to f in the Lp-norm,
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
and uses repeatedly that when p < 1.The case p = ∞ reduces to a simple question about uniform convergence outside a μ–negligible set.