Titchmarsh convolution theorem
E. C. Titchmarsh proved the following theorem, known as the Titchmarsh convolution theorem, in 1926:
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
A corollary follows:
If the integral above is 0 for all then either or is almost everywhere 0 in the interval
The theorem can be restated in the following form:
- Let . Then if the right-hand side is finite.
- Similarly, if the right-hand side is finite.
This theorem essentially states that the well-known inclusion
is sharp at the boundary.
- If , then
The theorem lacks an elementary proof. 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).
- 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)
|url=(help). 232: 1530–1532, 1622–1624.
- Mikusiński, J. and Świerczkowski, S. (1960). "Titchmarsh's theorem on convolution and the theory of Dufresnoy". Prace Matematyczne. 4: 59–76.CS1 maint: uses authors parameter (link)