#jsDisabledContent { display:none; } My Account |  Register |  Help

# Titchmarsh convolution theorem

Article Id: WHEBN0008195500
Reproduction Date:

 Title: Titchmarsh convolution theorem Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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 in 1926:

If \phi\,(t) and \psi(t)\, are integrable functions, such that
\int_{0}^{x}\phi(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 \phi(t)=0\, almost everywhere in (0,\lambda)\,, and \psi(t)=0\, almost everywhere in (0,\mu)\,.

This result, known as the Titchmarsh convolution theorem, could be restated in the following form:

Let \phi,\,\psi\in L^1(\mathbb{R}). Then \inf\mathop{\rm supp}\,\phi\ast \psi =\inf\mathop{\rm supp}\,\phi+\inf\mathop{\rm supp}\,\psi if the right-hand side is finite.
Similarly, \sup\mathop{\rm supp}\,\phi\ast\psi=\sup\mathop{\rm supp}\,\phi+\sup\mathop{\rm supp}\,\psi if the right-hand side is finite.

This theorem essentially states that the well-known inclusion

{\rm supp}\,\phi\ast \psi \subset \mathop{\rm supp}\,\phi +\mathop{\rm 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 \phi,\,\psi\in\mathcal{E}'(\mathbb{R}^n), then \mathop{c.h.}\mathop{\rm supp}\,\phi\ast \psi=\mathop{c.h.}\mathop{\rm supp}\,\phi+\mathop{c.h.}\mathop{\rm supp}\,\psi.

Above, \mathop{c.h.} denotes the convex hull of the set. \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, Theorem of Carleman, and Theorem of Valiron. More proofs are contained in [Hörmander, Theorem 4.3.3] (harmonic analysis style), [Yosida, Chapter VI] (real analysis style), and [Levin, Lecture 16] (complex analysis style).

## References

• Lions, J.-L. (1951). "Supports de produits de composition".
• Mikusiński, J. and Świerczkowski, S. (1960). "Titchmarsh's theorem on convolution and the theory of Dufresnoy".
• Yosida, K. (1980). Functional Analysis. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 (6th ed.). Berlin: Springer-Verlag.
• Levin, B. Ya. (1996). Lectures on Entire Functions. Translations of Mathematical Monographs, vol. 150. Providence, RI: American Mathematical Society.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.