Titchmarsh convolution theorem

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.

Titchmarsh convolution theorem

If [math]\displaystyle{ \varphi(t)\, }[/math] and [math]\displaystyle{ \psi(t) }[/math] are integrable functions, such that

[math]\displaystyle{ \varphi * \psi = \int_0^x \varphi(t)\psi(x-t)\,dt=0 }[/math]

almost everywhere in the interval [math]\displaystyle{ 0\lt x\lt \kappa\, }[/math], then there exist [math]\displaystyle{ \lambda\geq0 }[/math] and [math]\displaystyle{ \mu\geq0 }[/math] satisfying [math]\displaystyle{ \lambda+\mu\ge\kappa }[/math] such that [math]\displaystyle{ \varphi(t)=0\, }[/math] almost everywhere in [math]\displaystyle{ 0\lt t\lt \lambda }[/math] and [math]\displaystyle{ \psi(t)=0\, }[/math] almost everywhere in [math]\displaystyle{ 0\lt t\lt \mu. }[/math]

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

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

The theorem can be restated in the following form:

Let [math]\displaystyle{ \varphi, \psi\in L^1(\mathbb{R}) }[/math]. Then [math]\displaystyle{ \inf\operatorname{supp} \varphi\ast \psi=\inf\operatorname{supp} \varphi+\inf\operatorname{supp} \psi }[/math] if the left-hand side is finite. Similarly, [math]\displaystyle{ \sup\operatorname{supp} \varphi\ast\psi = \sup\operatorname{supp}\varphi + \sup\operatorname{supp} \psi }[/math] if the right-hand side is finite.

Above, [math]\displaystyle{ \operatorname{supp} }[/math] denotes the support of a function f (i.e., the closure of the complement of f^-1(0)) and [math]\displaystyle{ \inf }[/math] and [math]\displaystyle{ \sup }[/math] denote the infimum and supremum. This theorem essentially states that the well-known inclusion [math]\displaystyle{ \operatorname{supp}\varphi\ast \psi \subset \operatorname{supp}\varphi+\operatorname{supp}\psi }[/math] 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:

If [math]\displaystyle{ \varphi, \psi\in\mathcal{E}'(\mathbb{R}^n) }[/math], then [math]\displaystyle{ \operatorname{c.h.} \operatorname{supp} \varphi\ast \psi=\operatorname{c.h.} \operatorname{supp} \varphi+\operatorname{c.h.}\operatorname{supp} \psi }[/math]

Above, [math]\displaystyle{ \operatorname{c.h.} }[/math] denotes the convex hull of the set and [math]\displaystyle{ \mathcal{E}' (\mathbb{R}^n) }[/math] 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 or complex-variable 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.