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| The '''Titchmarsh convolution theorem''' is named after [[Edward Charles Titchmarsh]],
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| a British mathematician. The theorem describes the properties of the [[support (mathematics)|support]] of the [[convolution]] of two functions.
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| == Titchmarsh convolution theorem ==
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| [[Edward Charles Titchmarsh|E.C. Titchmarsh]] proved the following theorem in 1926:
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| :If <math>\phi\,(t)</math> and <math>\psi(t)\,</math> are integrable functions, such that | |
| ::<math>\int_{0}^{x}\phi(t)\psi(x-t)\,dt=0</math>
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| :[[almost everywhere]] in the interval <math>0<x<\kappa\,</math>, then there exist <math>\lambda\geq0</math> and <math>\mu\geq0</math> satisfying <math>\lambda+\mu\ge\kappa</math> such that <math>\phi(t)=0\,</math> almost everywhere in <math>(0,\lambda)\,</math>, and <math>\psi(t)=0\,</math> almost everywhere in <math>(0,\mu)\,</math>.
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| This result, known as the Titchmarsh convolution theorem, could be restated in the following form:
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| :Let <math>\phi,\,\psi\in L^1(\mathbb{R})</math>. Then <math>\inf\mathop{\rm supp}\,\phi\ast \psi
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| =\inf\mathop{\rm supp}\,\phi+\inf\mathop{\rm supp}\,\psi</math> if the right-hand side is finite.
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| :Similarly, <math>\sup\mathop{\rm supp}\,\phi\ast\psi=\sup\mathop{\rm supp}\,\phi+\sup\mathop{\rm supp}\,\psi</math> if the right-hand side is finite.
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| This theorem essentially states that the well-known inclusion
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| :<math>
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| {\rm supp}\,\phi\ast \psi
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| \subset
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| \mathop{\rm supp}\,\phi
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| +\mathop{\rm supp}\,\psi
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| </math>
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| is sharp at the boundary.
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| The higher-dimensional generalization in terms of the
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| [[convex hull]] of the supports was proved by
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| [[Jacques-Louis Lions|J.-L. Lions]] in 1951:
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| : ''If <math>\phi,\,\psi\in\mathcal{E}'(\mathbb{R}^n)</math>, then <math>\mathop{c.h.}\mathop{\rm supp}\,\phi\ast \psi=\mathop{c.h.}\mathop{\rm supp}\,\phi+\mathop{c.h.}\mathop{\rm supp}\,\psi.</math>''
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| Above, <math>\mathop{c.h.}</math> denotes the [[convex hull]] of the set.
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| <math>\mathcal{E}'(\mathbb{R}^n)</math> | |
| denotes
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| the space of [[distribution (mathematics)|distributions]] with [[compact support]].
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| The theorem lacks an '''elementary''' proof.
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| The original proof by Titchmarsh
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| is based on the [[Phragmén–Lindelöf principle]],
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| [[Jensen's inequality]],
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| [[Theorem of Carleman]],
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| and
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| [[Bloch's theorem (complex variables)#Valiron's theorem|Theorem of Valiron]].
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| More proofs are contained in [Hörmander, Theorem 4.3.3] ([[harmonic analysis]] style),
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| [Yosida, Chapter VI] ([[real analysis]] style),
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| and [Levin, Lecture 16] ([[complex analysis]] style).
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| ==References==
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| *{{cite journal
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| | author = Titchmarsh, E.C.
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| | authorlink = Edward Charles Titchmarsh
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| | title = The zeros of certain integral functions
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| | journal = [[Proceedings of the London Mathematical Society]]
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| | volume = 25
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| | year = 1926
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| | pages = 283–302
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| | doi = 10.1112/plms/s2-25.1.283}}
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| *{{cite journal
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| | author = Lions, J.-L.
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| | title = Supports de produits de composition
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| | format = I and II
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| | journal = [[Les Comptes rendus de l'Académie des sciences]]
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| | volume = 232
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| | year = 1951
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| | pages = 1530–1532, 1622–1624}}
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| *{{cite journal
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| | authors = Mikusiński, J. and Świerczkowski, S.
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| | title = Titchmarsh's theorem on convolution and the theory of Dufresnoy
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| | journal = [[Prace Matematyczne]]
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| | volume = 4
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| | year = 1960
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| | pages = 59-76}}
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| *{{cite book
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| | author = Yosida, K.
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| | title = Functional Analysis
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| | edition = 6th
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| | series = Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123
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| | publisher = Springer-Verlag
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| | location = Berlin
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| | year = 1980}}
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| *{{cite book
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| | authorlink = Lars Hörmander
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| | author = Hörmander, L.
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| | title = The Analysis of Linear Partial Differential Operators, I
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| | edition = 2nd
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| | series = Springer Study Edition
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| | publisher = Springer-Verlag
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| | location = Berlin
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| | year = 1990}}
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| *{{cite book
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| | author = Levin, B. Ya.
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| | title = Lectures on Entire Functions
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| | series = Translations of Mathematical Monographs, vol. 150
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| | publisher = American Mathematical Society
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| | location = Providence, RI
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| | year = 1996}}
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| [[Category:Theorems in harmonic analysis]]
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| [[Category:Theorems in complex analysis]]
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| [[Category:Theorems in real analysis]]
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