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

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

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

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.

The theorem can be restated in the following form:

Let $\varphi ,\psi \in L^{1}(\mathbb {R} )$. Then $\inf \operatorname {supp} \varphi \ast \psi =\inf \operatorname {supp} \varphi +\inf \operatorname {supp} \psi$ if the right-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.

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 J.-L. Lions in 1951:

If $\varphi ,\psi \in {\mathcal {E}}'(\mathbb {R} ^{n})$, 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 ${\mathcal {E}}'(\mathbb {R} ^{n})$ 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, Carleman's theorem, and Valiron's theorem.

More proofs are contained in:

• "Theorem 4.3.3", Hörmander, L. (1990). The Analysis of Linear Partial Differential Operators, I. Springer Study Edition (2nd ed.). Berlin: Springer-Verlag.

(harmonic analysis style)

• "Chapter VI", Yosida, K. (1980). Functional Analysis. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 (6th ed.). Berlin: Springer-Verlag.

(real analysis style)

• "Lecture 16", Levin, B. Ya. (1996). Lectures on Entire Functions. Translations of Mathematical Monographs, vol. 150. Providence, RI: American Mathematical Society.

(complex analysis style).

Bibliography

• Titchmarsh, E.C. (1926). "The zeros of certain integral functions". Proceedings of the London Mathematical Society. 25: 283–302. doi:10.1112/plms/s2-25.1.283.

• Lions, J.-L. (1951). "Supports de produits de composition". Les Comptes rendus de l'Académie des sciences. 232: 1530–1532, 1622–1624. {{cite journal}}: |format= requires |url= (help)

• "Titchmarsh's theorem on convolution and the theory of Dufresnoy". Prace Matematyczne. 4: 59–76. 1960. {{cite journal}}: Unknown parameter |authors= ignored (help)