Titchmarsh convolution theorem explained

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 $\backslash varphi(t)\backslash ,$ and $\backslash psi(t)$ are integrable functions, such that

\varphi * \psi = \int ^{x} _{0} \varphi(t)\psi(x - t)dt =0 almost everywhere in the interval 0 <x<\kappa , then there exist λ \geq 0 and \mu\geq 0 satisfying λ+ \mu\ge\kappa such that \varphi(t) =0 almost everywhere in 0 <t< λ and \psi(t) =0 almost everywhere in 0 <t<\mu.

As a corollary, if the integral above is 0 for all $x>0,$ then either $\backslash varphi\backslash ,$ or $\backslash psi$ is almost everywhere 0 in the interval 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(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 l{E}'( 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 l{E}' ( 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]

Notes and References

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