Titchmarsh convolution theorem
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
top
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.cite-ref-1[1]
Contents
β’ References
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
Titchmarsh convolution theorem
If Ο ( t ) {\textstyle \varphi (t)\,} and Ο ( t ) {\textstyle \psi (t)} are integrable functions, such that
Ο β Ο = β« 0 x Ο ( t ) Ο ( x β t ) d t = 0 {\displaystyle \varphi *\psi =\int _{0}^{x}\varphi (t)\psi (x-t)\,dt=0}
almost everywhere in the interval 0 < x < ΞΊ {\displaystyle 0<x<\kappa \,} , then there exist Ξ» β₯ 0 {\displaystyle \lambda \geq 0} and ΞΌ β₯ 0 {\displaystyle \mu \geq 0} satisfying Ξ» + ΞΌ β₯ ΞΊ {\displaystyle \lambda +\mu \geq \kappa } such that Ο ( t ) = 0 {\displaystyle \varphi (t)=0\,} almost everywhere in 0 < t < Ξ» {\displaystyle 0<t<\lambda } and Ο ( t ) = 0 {\displaystyle \psi (t)=0\,} almost everywhere in 0 < t < ΞΌ . {\displaystyle 0<t<\mu .}
As a corollary, if the integral above is 0 for all x > 0 , {\textstyle x>0,} then either Ο {\textstyle \varphi \,} or Ο {\textstyle \psi } is almost everywhere 0 in the interval [ 0 , + β ) . {\textstyle [0,+\infty ).} Thus the convolution of two functions on [ 0 , + β ) {\textstyle [0,+\infty )} cannot be identically zero unless at least one of the two functions is identically zero.
As another corollary, if Ο β Ο ( x ) = 0 {\displaystyle \varphi *\psi (x)=0} for all x β [ 0 , ΞΊ ] {\displaystyle x\in [0,\kappa ]} and one of the function Ο {\displaystyle \varphi } or Ο {\displaystyle \psi } is almost everywhere not null in this interval, then the other function must be null almost everywhere in [ 0 , ΞΊ ] {\displaystyle [0,\kappa ]} .
The theorem can be restated in the following form:
Let Ο , Ο β L 1 ( R ) {\displaystyle \varphi ,\psi \in L^{1}(\mathbb {R} )} . Then inf supp β‘ Ο β Ο = inf supp β‘ Ο + inf supp β‘ Ο {\displaystyle \inf \operatorname {supp} \varphi \ast \psi =\inf \operatorname {supp} \varphi +\inf \operatorname {supp} \psi } if the left-hand side is finite. Similarly, sup supp β‘ Ο β Ο = sup supp β‘ Ο + sup supp β‘ Ο {\displaystyle \sup \operatorname {supp} \varphi \ast \psi =\sup \operatorname {supp} \varphi +\sup \operatorname {supp} \psi } if the right-hand side is finite.
Above, supp {\displaystyle \operatorname {supp} } denotes the support of a function f (i.e., the closure of the complement of fβ1(0)) and inf {\displaystyle \inf } and sup {\displaystyle \sup } denote the infimum and supremum. This theorem essentially states that the well-known inclusion supp β‘ Ο β Ο β supp β‘ Ο + supp β‘ Ο {\displaystyle \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:cite-ref-2[2]
If Ο , Ο β E β² ( R n ) {\displaystyle \varphi ,\psi \in {\mathcal {E}}'(\mathbb {R} ^{n})} , then c . h . β‘ supp β‘ Ο β Ο = c . h . β‘ supp β‘ Ο + c . h . β‘ supp β‘ Ο {\displaystyle \operatorname {c.h.} \operatorname {supp} \varphi \ast \psi =\operatorname {c.h.} \operatorname {supp} \varphi +\operatorname {c.h.} \operatorname {supp} \psi }
Above, c . h . {\displaystyle \operatorname {c.h.} } denotes the convex hull of the set and E β² ( R n ) {\displaystyle {\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 PhragmΓ©nβLindelΓΆf principle, Jensen's inequality, Carleman's theorem, and Valiron's theorem. The theorem has since been proven several more times, typically using either real-variablecite-ref-3[3]cite-ref-4[4]cite-ref-5[5] or complex-variablecite-ref-6[6]cite-ref-7[7]cite-ref-8[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.cite-ref-9[9]
References
cite-note-22. β citereflions1951Lions, Jacques-Louis (1951). "Supports de produits de composition". Comptes rendus. 232 (17): 1530β1532.
cite-note-33. β citerefdoss1988Doss, Raouf (1988). "An elementary proof of Titchmarsh's convolution theorem" (PDF). Proceedings of the American Mathematical Society. 104 (1).
cite-note-77. β citerefdufresnoy1947Dufresnoy, Jacques (1947). "Sur le produit de composition de deux fonctions". Comptes rendus. 225: 857β859.