Titchmarsh convolution theorem
Titchmarsh convolution theorem 
E.C. Titchmarsh proved the following 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 .
This result, known as the Titchmarsh convolution theorem, could 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, 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).
- 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 ) 232: 1530–1532, 1622–1624.
- Yosida, K. (1980). Functional Analysis. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 (6th ed.). Berlin: Springer-Verlag.
- Hörmander, L. (1990). The Analysis of Linear Partial Differential Operators, I. Springer Study Edition (2nd ed.). Berlin: Springer-Verlag.
- Levin, B. Ya. (1996). Lectures on Entire Functions. Translations of Mathematical Monographs, vol. 150. Providence, RI: American Mathematical Society.