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.^[1]

Titchmarsh convolution theorem

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

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

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

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

The theorem can be restated in the following form:

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

Above, ${\displaystyle \operatorname {supp} }$ denotes the support of a function f (i.e., the closure of the complement of f^-1(0)) and ${\displaystyle \inf }$ and ${\displaystyle \sup }$ denote the infimum and supremum. This theorem essentially states that the well-known inclusion ${\displaystyle \operatorname {supp} \varphi \ast \psi \subset \operatorname {supp} \varphi +\operatorname {supp} \psi }$ 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 ${\displaystyle \varphi ,\psi \in {\mathcal {E}}'(\mathbb {R} ^{n})}$, then ${\displaystyle \operatorname {c.h.} \operatorname {supp} \varphi \ast \psi =\operatorname {c.h.} \operatorname {supp} \varphi +\operatorname {c.h.} \operatorname {supp} \psi }$

Above, ${\displaystyle \operatorname {c.h.} }$ denotes the convex hull of the set and ${\displaystyle {\mathcal {E}}'(\mathbb {R} ^{n})}$ 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]

References