If \varphi(t)\, and \psi(t) are integrable functions, such that :\varphi * \psi = \int_0^x \varphi(t)\psi(x-t)\,dt=0
almost everywhere in the interval 0, then there exist \lambda\geq0 and \mu\geq0 satisfying \lambda+\mu\ge\kappa such that \varphi(t)=0\, almost everywhere in 0 and \psi(t)=0\, almost everywhere in 0 As a corollary, if the integral above is 0 for all x>0, then either \varphi\, or \psi is almost everywhere 0 in the interval [0,+\infty). Thus the convolution of two functions on [0,+\infty) cannot be identically zero unless at least one of the two functions is identically zero. As another corollary, if \varphi * \psi (x) = 0 for all x\in [0, \kappa] and one of the function \varphi or \psi is almost everywhere not null in this interval, then the other function must be null almost everywhere in [0,\kappa]. The theorem can be restated in the following form: :Let \varphi, \psi\in L^1(\mathbb{R}). Then \inf\operatorname{supp} \varphi\ast \psi=\inf\operatorname{supp} \varphi+\inf\operatorname{supp} \psi if the left-hand side is finite. Similarly, \sup\operatorname{supp} \varphi\ast\psi = \sup\operatorname{supp}\varphi + \sup\operatorname{supp} \psi if the right-hand side is finite. Above, \operatorname{supp} denotes the support of a function f (i.e., the closure of the complement of f−1(0)) and \inf and \sup denote the
infimum and supremum. This theorem essentially states that the well-known inclusion \operatorname{supp}\varphi\ast \psi \subset \operatorname{supp}\varphi+\operatorname{supp}\psi 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: :If \varphi, \psi\in\mathcal{E}'(\mathbb{R}^n), then \operatorname{c.h.} \operatorname{supp} \varphi\ast \psi=\operatorname{c.h.} \operatorname{supp} \varphi+\operatorname{c.h.}\operatorname{supp} \psi Above, \operatorname{c.h.} denotes the
convex hull of the set and \mathcal{E}' (\mathbb{R}^n) 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. ==References==