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.
are integrable functions, such that almost everywhere in the interval
λ + μ ≥ κ
As a corollary, if the integral above is 0 for all
Thus the convolution of two functions on
φ ∗ ψ ( x ) = 0
The theorem can be restated in the following form: Above,
denotes the support of a function f (i.e., the closure of the complement of f−1(0)) and
denote the infimum and supremum.
This theorem essentially states that the well-known inclusion
supp φ ∗ ψ ⊂ supp φ + supp ψ
The higher-dimensional generalization in terms of the convex hull of the supports was proven by Jacques-Louis Lions in 1951:[2] Above,
denotes the convex hull of the set and
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[3][4][5] or complex-variable[6][7][8] 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.