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 \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<x<\kappa\,, then there exist \lambda\geq0 and \mu\geq0 satisfying \lambda+\mu\ge\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 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:[2]

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[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

  1. ^ Titchmarsh, E. C. (1926). "The Zeros of Certain Integral Functions". Proceedings of the London Mathematical Society. s2-25 (1): 283–302. doi:10.1112/plms/s2-25.1.283
  2. ^ Lions, Jacques-Louis (1951). "Supports de produits de composition". Comptes rendus. 232 (17): 1530–1532.
  3. ^ Doss, Raouf (1988). "An elementary proof of Titchmarsh's convolution theorem". Proceedings of the American Mathematical Society. 104 (1)
  4. ^ Kalisch, G. K. (1962-10-01). "A functional analysis proof of titchmarsh's theorem on convolution". Journal of Mathematical Analysis and Applications. 5 (2): 176–183. doi:10.1016/S0022-247X(62)80002-X. ISSN 0022-247X
  5. ^ Mikusiński, J. (1953). "A new proof of Titchmarsh's theorem on convolution". Studia Mathematica. 13 (1): 56–58. doi:10.4064/sm-13-1-56-58. ISSN 0039-3223
  6. ^ Crum, M. M. (1941). "On the resultant of two functions". The Quarterly Journal of Mathematics. os-12 (1): 108–111. doi:10.1093/qmath/os-12.1.108. ISSN 0033-5606
  7. ^ Dufresnoy, Jacques (1947). "Sur le produit de composition de deux fonctions". Comptes rendus. 225: 857–859.
  8. ^ Boas, Ralph P. (1954). Entire functions. New York: Academic Press. ISBN 0-12-108150-8. OCLC 847696
  9. ^ Rota, Gian-Carlo (1998-06-01). "Ten Mathematics Problems I will never solve" (in German). Mitteilungen der Deutschen Mathematiker-Vereinigung. 6 (2): 45–52. doi:10.1515/dmvm-1998-0215. ISSN 0942-5977. S2CID 120569917