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.

Titchmarsh convolution theorem

are integrable functions, such that

almost everywhere

cannot be identically zero unless at least one of the two functions is identically zero.

.

The theorem can be restated in the following form:

1. if the right-hand side is finite.

denotes the support of a function f (i.e., the closure of the complement of f−1 denote the infimum and supremum 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:

denotes the convex hull 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 or complex-variable 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.

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