Titchmarsh convolution theorem

The Titchmarsh convolution theorem is named after Edward Charles Titchmarsh, a British mathematician. The theorem describes the properties of the support of the convolution of two functions.

Titchmarsh convolution theorem[]

E. C. Titchmarsh proved the following theorem, known as the Titchmarsh convolution theorem, in 1926:

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

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

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 ${\displaystyle 0<t<\mu .}$

A corollary follows:

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

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 proved by J.-L. Lions in 1951:

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. ${\displaystyle {\mathcal {E}}'(\mathbb {R} ^{n})}$ denotes the space of distributions with compact support.

The theorem lacks an elementary proof.^[1] The original proof by Titchmarsh is based on the Phragmén–Lindelöf principle, Jensen's inequality, the Theorem of Carleman, and Theorem of Valiron. 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 (I and II) |format= requires |url= (help). 232: 1530–1532, 1622–1624.

• Mikusiński, J. and Świerczkowski, S. (1960). "Titchmarsh's theorem on convolution and the theory of Dufresnoy". . 4: 59–76.CS1 maint: uses authors parameter (link)

References[]