Titchmarsh convolution theorem

The Titchmarsh convolution theoremis named after Edward Charles Titchmarsh.The theorem describes the properties of the
support of the convolutionof two functions.

Titchmarsh convolution theorem

E.C. Titchmarshproved the following theorem in 1926:

:If $phi(t)$ and $psi(t)$ are integrable functions, such that::$int\_\{0\}^\{x\}phi(t)psi(x-t),dt=0$:almost everywhere in the interval

This result, known as the Titchmarsh convolution theorem,could be restated in the following form:

:Let $phi,,psiin\; L^1(mathbb\{R\})$. Then $infmathop\{\; m\; supp\},phiast\; psi=infmathop\{\; m\; supp\},phi+infmathop\{\; m\; supp\},psi$ if the right-hand side is finite.:Similarly, $supmathop\{\; m\; supp\},phiastpsi=supmathop\{\; m\; supp\},phi+supmathop\{\; m\; supp\},psi$ if the right-hand side is finite.

This theorem essentially states that the well-known inclusion:$\{\; m\; supp\},phiast\; psisubsetmathop\{\; m\; supp\},phi+mathop\{\; m\; supp\},psi$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:

: "If $phi,,psiinmathcal\{E\}\text{'}(mathbb\{R\}^n)$, then $mathop\{c.h.\}mathop\{\; m\; supp\},phiast\; psi=mathop\{c.h.\}mathop\{\; m\; supp\},phi+mathop\{c.h.\}mathop\{\; m\; supp\},psi$."

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

The theorem lacks an elementary proof.The original proof by Titchmarshis 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).

References

*cite journal
author = Titchmarsh, E.C.
authorlink = Edward Charles Titchmarsh
title = The zeros of certain integral functions
journal = Proceedings of the London Mathematical Society
volume = 25
year = 1926
pages = 283–302
doi = 10.1112/plms/s2-25.1.283

*cite journal
author = Lions, J.-L.
title = Supports de produits de composition
format = I and II
journal = Les Comptes rendus de l'Académie des sciences
volume = 232
year = 1951
pages = 1530–1532, 1622–1624

*cite book
author = Yosida, K.
title = Functional Analysis
edition = 6th ed.
series = Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123
publisher = Springer-Verlag
location = Berlin
year = 1980

*cite book
authorlink = Lars Hörmander
author = Hörmander, L.
title = The Analysis of Linear Partial Differential Operators, I
edition = 2nd ed.
series = Springer Study Edition
publisher = Springer-Verlag
location = Berlin
year = 1990

*cite book
author = Levin, B. Ya.
title = Lectures on Entire Functions
series = Translations of Mathematical Monographs, vol. 150
publisher = American Mathematical Society
location = Providence, RI
year = 1996