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.{{Cite journal |last=Titchmarsh |first=E. C. |date=1926 |title=The Zeros of Certain Integral Functions |url=http://doi.wiley.com/10.1112/plms/s2-25.1.283 |journal=Proceedings of the London Mathematical Society |language=en |volume=s2-25 |issue=1 |pages=283–302 |doi=10.1112/plms/s2-25.1.283}}

Titchmarsh convolution theorem

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

: $\varphi * \psi = \int_0^x \varphi(t)\psi(x-t)\,dt=0$

almost everywhere in the interval $0, then there exist \lambda\geq0 and \mu\geq0 satisfying \lambda+\mu\ge\kappa such that \varphi(t)=0\, almost everywhere in 0 and \psi(t)=0\, almost everywhere in 0$

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

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

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 left-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.

Above, $\operatorname{supp}$ denotes the support of a function f (i.e., the closure of the complement of f⁻¹(0)) and $\inf$ and $\sup$ denote the infimum and supremum. 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 Jacques-Louis Lions in 1951:{{Cite journal |last=Lions |first=Jacques-Louis |date=1951 |title=Supports de produits de composition |journal=Comptes rendus |volume=232 |issue=17 |pages=1530–1532}}

: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 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{{Cite journal |last=Doss |first=Raouf |date=1988 |title=An elementary proof of Titchmarsh's convolution theorem |url=https://www.ams.org/journals/proc/1988-104-01/S0002-9939-1988-0958063-5/S0002-9939-1988-0958063-5.pdf |journal=Proceedings of the American Mathematical Society |volume=104 |issue=1}}{{Cite journal |last=Kalisch |first=G. K. |date=1962-10-01 |title=A functional analysis proof of titchmarsh's theorem on convolution |journal=Journal of Mathematical Analysis and Applications |language=en |volume=5 |issue=2 |pages=176–183 |doi=10.1016/S0022-247X(62)80002-X |issn=0022-247X|doi-access=free }}{{Cite journal |last=Mikusiński |first=J. |date=1953 |title=A new proof of Titchmarsh's theorem on convolution |journal=Studia Mathematica |language=en |volume=13 |issue=1 |pages=56–58 |doi=10.4064/sm-13-1-56-58 |issn=0039-3223|doi-access=free }} or complex-variable{{Cite journal |last=Crum |first=M. M. |date=1941 |title=On the resultant of two functions |url=https://academic.oup.com/qjmath/article-lookup/doi/10.1093/qmath/os-12.1.108 |journal=The Quarterly Journal of Mathematics |language=en |volume=os-12 |issue=1 |pages=108–111 |doi=10.1093/qmath/os-12.1.108 |issn=0033-5606}}{{Cite journal |last=Dufresnoy |first=Jacques |date=1947 |title=Sur le produit de composition de deux fonctions |journal=Comptes rendus |volume=225 |pages=857–859}}{{Cite book |last=Boas |first=Ralph P. |url=https://www.worldcat.org/oclc/847696 |title=Entire functions |date=1954 |publisher=Academic Press |isbn=0-12-108150-8 |location=New York |oclc=847696}} 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.{{Cite journal |last=Rota |first=Gian-Carlo |date=1998-06-01 |title=Ten Mathematics Problems I will never solve |journal=Mitteilungen der Deutschen Mathematiker-Vereinigung |language=de |volume=6 |issue=2 |pages=45–52 |doi=10.1515/dmvm-1998-0215 |s2cid=120569917 |issn=0942-5977|doi-access=free }}

References

Category:Theorems in harmonic analysis

Category:Theorems in complex analysis

Category:Theorems in real analysis