Titchmarsh convolution theorem
Encyclopedia
The Titchmarsh convolution theorem is named after Edward Charles Titchmarsh
,
a British mathematician. The theorem describes the properties of the support
of the convolution
of two functions.
proved the following theorem in 1926:
This result, known as the Titchmarsh convolution theorem, could be restated in the following form:
This theorem essentially states that the well-known inclusion
is sharp at the boundary.
The higher-dimensional generalization in terms of the
convex hull
of the supports was proved by
J.-L. Lions
in 1951:
Above,
denotes the convex hull
of the set.
denotes
the space of distributions
with compact support.
The theorem lacks an elementary proof.
The original proof by Titchmarsh
is based on the Phragmén–Lindelöf principle,
Jensen's inequality
,
Theorem of Carleman,
and
Theorem of Valiron.
More proofs are contained in [Hörmander, Theorem 4.3.3] (harmonic analysis
style),
[Yosida, Chapter VI] (real analysis
style),
and [Levin, Lecture 16] (complex analysis
style).
Edward Charles Titchmarsh
Edward Charles "Ted" Titchmarsh was a leading British mathematician.He was educated at King Edward VII School and Balliol College, Oxford, where he began his studies in October 1917....
,
a British mathematician. The theorem describes the properties of the support
Support (mathematics)
In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
of the convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
of two functions.
Titchmarsh convolution theorem
E.C. TitchmarshEdward Charles Titchmarsh
Edward Charles "Ted" Titchmarsh was a leading British mathematician.He was educated at King Edward VII School and Balliol College, Oxford, where he began his studies in October 1917....
proved the following theorem in 1926:
- If
and 
are integrable functions, such that
- almost everywhereAlmost everywhereIn measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
in the interval
, then there exist 
and 
satisfying 
such that 
almost everywhere in 
, and 
almost everywhere in 
.
This result, known as the Titchmarsh convolution theorem, could be restated in the following form:
- Let
. Then 
if the right-hand side is finite. - Similarly,
if the right-hand side is finite.
This theorem essentially states that the well-known inclusion
is sharp at the boundary.
The higher-dimensional generalization in terms of the
convex hull
Convex hull
In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
of the supports was proved by
J.-L. Lions
Jacques-Louis Lions
Jacques-Louis Lions ForMemRS was a French mathematician who made contributions to the theory of partial differential equations and to stochastic control, among other areas. He received the SIAM's John Von Neumann prize in 1986. Lions is listed as an ISI highly cited researcher.-Biography:After...
in 1951:
- If
, then 
Above,
denotes the convex hullConvex hull
In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
of the set.
denotes
the space of distributions
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. 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...
with compact support.
The theorem lacks an elementary proof.
The original proof by Titchmarsh
is based on the Phragmén–Lindelöf principle,
Jensen's inequality
Jensen's inequality
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. Given its generality, the inequality appears in many forms depending on the context,...
,
Theorem of Carleman,
and
Theorem of Valiron.
More proofs are contained in [Hörmander, Theorem 4.3.3] (harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
style),
[Yosida, Chapter VI] (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...
style),
and [Levin, Lecture 16] (complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
style).