Titchmarsh convolution theorem

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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 and are integrable functions, such that

almost everywhere in the interval , then there exist and satisfying such that almost everywhere in and almost everywhere in

As a corollary, if the integral above is 0 for all then either or is almost everywhere 0 in the interval Thus the convolution of two functions on cannot be identically zero unless at least one of the two functions is identically zero.

As another corollary, if for all and one of the function or is almost everywhere not null in this interval, then the other function must be null almost everywhere in .

The theorem can be restated in the following form:

Let . Then if the left-hand side is finite. Similarly, if the right-hand side is finite.

Above, denotes the support of a function f (i.e., the closure of the complement of f-1(0)) and and denote the infimum and supremum. This theorem essentially states that the well-known inclusion 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 , then

Above, denotes the convex hull of the set and 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]

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  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" (PDF). 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.
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  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". Mitteilungen der Deutschen Mathematiker-Vereinigung (in German). 6 (2): 45–52. doi: 10.1515/dmvm-1998-0215 . ISSN   0942-5977. S2CID   120569917.