Titchmarsh convolution theorem


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

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:
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:
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 , , and .