Titchmarsh convolution theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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, 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.

The higher-dimensional generalization in terms of the convex hull of the supports was proved by J.-L. Lions in 1951:

If , then

Above, denotes the convex hull of the set 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, the 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).


  • Mikusiński, J. and Świerczkowski, S. (1960). "Titchmarsh's theorem on convolution and the theory of Dufresnoy". Prace Matematyczne. 4: 59–76. 
  • 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.