Titchmarsh convolution theorem

Last updated January 20, 2025

The Titchmarsh convolution theorem describes the properties of the support of the convolution of two functions. It was proven by Edward Charles Titchmarsh in 1926.^[1]

Titchmarsh convolution theorem

If ${\textstyle \varphi (t)\,}$ and ${\textstyle \psi (t)}$ are integrable functions, such that

\varphi *\psi =\int _{0}^{x}\varphi (t)\psi (x-t)\,dt=0

almost everywhere in the interval $0<x<\kappa \,$ , then there exist $\lambda \geq 0$ and $\mu \geq 0$ satisfying $\lambda +\mu \geq \kappa$ such that $\varphi (t)=0\,$ almost everywhere in $0<t<\lambda$ and $\psi (t)=0\,$ almost everywhere in $0<t<\mu .$

As a corollary, if the integral above is 0 for all ${\textstyle x>0,}$ then either ${\textstyle \varphi \,}$ or ${\textstyle \psi }$ is almost everywhere 0 in the interval ${\textstyle [0,+\infty ).}$ Thus the convolution of two functions on ${\textstyle [0,+\infty )}$ cannot be identically zero unless at least one of the two functions is identically zero.

As another corollary, if $\varphi *\psi (x)=0$ for all $x\in [0,\kappa ]$ and one of the function $\varphi$ or $\psi$ is almost everywhere not null in this interval, then the other function must be null almost everywhere in $[0,\kappa ]$ .

The theorem can be restated in the following form:

Let

\varphi ,\psi \in L^{1}(\mathbb {R} )

. Then

\inf \operatorname {supp} \varphi \ast \psi =\inf \operatorname {supp} \varphi +\inf \operatorname {supp} \psi

if the left-hand side is finite. Similarly,

\sup \operatorname {supp} \varphi \ast \psi =\sup \operatorname {supp} \varphi +\sup \operatorname {supp} \psi

if the right-hand side is finite.

Above, $\operatorname {supp}$ denotes the support of a function f (i.e., the closure of the complement of f⁻¹(0)) and $\inf$ and $\sup$ denote the infimum and supremum. This theorem essentially states that the well-known inclusion $\operatorname {supp} \varphi \ast \psi \subset \operatorname {supp} \varphi +\operatorname {supp} \psi$ is sharp at the boundary.

The higher-dimensional generalization in terms of the convex hull of the supports was proven by Jacques-Louis Lions in 1951:^[2]

If

\varphi ,\psi \in {\mathcal {E}}'(\mathbb {R} ^{n})

, then

\operatorname {c.h.} \operatorname {supp} \varphi \ast \psi =\operatorname {c.h.} \operatorname {supp} \varphi +\operatorname {c.h.} \operatorname {supp} \psi

Above, $\operatorname {c.h.}$ denotes the convex hull of the set and ${\mathcal {E}}'(\mathbb {R} ^{n})$ denotes the space of distributions with compact support.

The original proof by Titchmarsh uses complex-variable techniques, and is based on the Phragmén–Lindelöf principle, Jensen's inequality, Carleman's theorem, and Valiron's theorem. The theorem has since been proven several more times, typically using either real-variable ^[3]^[4]^[5] or complex-variable^[6]^[7]^[8] methods. Gian-Carlo Rota has stated that no proof yet addresses the theorem's underlying combinatorial structure, which he believes is necessary for complete understanding.^[9]

Related Research Articles

The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism.

In mathematics, specifically abstract algebra, the isomorphism theorems are theorems that describe the relationship among quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.

Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.

In mathematics, a self-adjoint operator on a complex vector space V with inner product $is a linear map A that is its own adjoint. That is, for all ∊ V . If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A * . By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.$

<span class="mw-page-title-main">Jensen's inequality</span> Theorem of convex functions

In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation.

In mathematics as well as physics, a linear operator $acting on an inner product space is called positive-semidefinite if, for every, and, where is the domain of . Positive-semidefinite operators are denoted as . The operator is said to be positive-definite, and written, if for all .$

In probability theory, the central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases to infinity. Under stronger assumptions, the Berry–Esseen theorem, or Berry–Esseen inequality, gives a more quantitative result, because it also specifies the rate at which this convergence takes place by giving a bound on the maximal error of approximation between the normal distribution and the true distribution of the scaled sample mean. The approximation is measured by the Kolmogorov–Smirnov distance. In the case of independent samples, the convergence rate is $n -1/2$ , where $n$ is the sample size, and the constant is estimated in terms of the third absolute normalized moment.

In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem.

In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an interval [a, b] ⊂ R, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation x ↦ f(x), for x ∈ [a, b]. Functions whose total variation is finite are called functions of bounded variation.

In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces. It can be viewed as the starting point of many results of similar nature.

In mathematics, mollifiers are particular smooth functions, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. Intuitively, given a (generalized) function, convolving it with a mollifier "mollifies" it, that is, its sharp features are smoothed, while still remaining close to the original.

In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.

In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra $over the real or complex numbers that at the same time is also a Fréchet space. The multiplication operation for is required to be jointly continuous. If is an increasing family of seminorms for the topology of, the joint continuity of multiplication is equivalent to there being a constant and integer for each such that for all . Fréchet algebras are also called B 0 -algebras.$

In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mathematician Gaspard Monge in 1781.

In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems.

In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.

In mathematical analysis, and especially in real, harmonic analysis and functional analysis, an Orlicz space is a type of function space which generalizes the L^p spaces. Like the L^p spaces, they are Banach spaces. The spaces are named for Władysław Orlicz, who was the first to define them in 1932.

In probability theory, a subgaussian distribution, the distribution of a subgaussian random variable, is a probability distribution with strong tail decay. More specifically, the tails of a subgaussian distribution are dominated by the tails of a Gaussian. This property gives subgaussian distributions their name.

In the mathematical subject geometric group theory, a fully irreducible automorphism of the free group F_n is an element of Out(F_n) which has no periodic conjugacy classes of proper free factors in F_n. Fully irreducible automorphisms are also referred to as "irreducible with irreducible powers" or "iwip" automorphisms. The notion of being fully irreducible provides a key Out(F_n) counterpart of the notion of a pseudo-Anosov element of the mapping class group of a finite type surface. Fully irreducibles play an important role in the study of structural properties of individual elements and of subgroups of Out(F_n).

In mathematics, Rathjen's $psi function$ is an ordinal collapsing function developed by Michael Rathjen. It collapses weakly Mahlo cardinals $to generate large countable ordinals. A weakly Mahlo cardinal is a cardinal such that the set of regular cardinals below is closed under . Rathjen uses this to diagonalise over the weakly inaccessible hierarchy.$

References

↑ Titchmarsh, E. C. (1926). "The Zeros of Certain Integral Functions". Proceedings of the London Mathematical Society. s2-25 (1): 283–302. doi:10.1112/plms/s2-25.1.283.
↑ Lions, Jacques-Louis (1951). "Supports de produits de composition". Comptes rendus . 232 (17): 1530–1532.
↑ Doss, Raouf (1988). "An elementary proof of Titchmarsh's convolution theorem" (PDF). Proceedings of the American Mathematical Society . 104 (1).
↑ Kalisch, G. K. (1962-10-01). "A functional analysis proof of titchmarsh's theorem on convolution". Journal of Mathematical Analysis and Applications. 5 (2): 176–183. doi: 10.1016/S0022-247X(62)80002-X . ISSN 0022-247X.
↑ Mikusiński, J. (1953). "A new proof of Titchmarsh's theorem on convolution". Studia Mathematica. 13 (1): 56–58. doi: 10.4064/sm-13-1-56-58 . ISSN 0039-3223.
↑ Crum, M. M. (1941). "On the resultant of two functions". The Quarterly Journal of Mathematics. os-12 (1): 108–111. doi:10.1093/qmath/os-12.1.108. ISSN 0033-5606.
↑ Dufresnoy, Jacques (1947). "Sur le produit de composition de deux fonctions". Comptes rendus . 225: 857–859.
↑ Boas, Ralph P. (1954). Entire functions. New York: Academic Press. ISBN 0-12-108150-8. OCLC 847696.
↑ Rota, Gian-Carlo (1998-06-01). "Ten Mathematics Problems I will never solve". Mitteilungen der Deutschen Mathematiker-Vereinigung (in German). 6 (2): 45–52. doi: 10.1515/dmvm-1998-0215 . ISSN 0942-5977. S2CID 120569917.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Titchmarsh, E. C. (1926). "The Zeros of Certain Integral Functions". Proceedings of the London Mathematical Society. s2-25 (1): 283–302. doi:10.1112/plms/s2-25.1.283.

[2] Lions, Jacques-Louis (1951). "Supports de produits de composition". Comptes rendus . 232 (17): 1530–1532.

[3] Doss, Raouf (1988). "An elementary proof of Titchmarsh's convolution theorem" (PDF). Proceedings of the American Mathematical Society . 104 (1).

[4] Kalisch, G. K. (1962-10-01). "A functional analysis proof of titchmarsh's theorem on convolution". Journal of Mathematical Analysis and Applications. 5 (2): 176–183. doi: 10.1016/S0022-247X(62)80002-X . ISSN 0022-247X.

[5] Mikusiński, J. (1953). "A new proof of Titchmarsh's theorem on convolution". Studia Mathematica. 13 (1): 56–58. doi: 10.4064/sm-13-1-56-58 . ISSN 0039-3223.

[6] Crum, M. M. (1941). "On the resultant of two functions". The Quarterly Journal of Mathematics. os-12 (1): 108–111. doi:10.1093/qmath/os-12.1.108. ISSN 0033-5606.

[7] Dufresnoy, Jacques (1947). "Sur le produit de composition de deux fonctions". Comptes rendus . 225: 857–859.

[8] Boas, Ralph P. (1954). Entire functions. New York: Academic Press. ISBN 0-12-108150-8. OCLC 847696.

[9] Rota, Gian-Carlo (1998-06-01). "Ten Mathematics Problems I will never solve". Mitteilungen der Deutschen Mathematiker-Vereinigung (in German). 6 (2): 45–52. doi: 10.1515/dmvm-1998-0215 . ISSN 0942-5977. S2CID 120569917.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]