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| The '''Titchmarsh convolution theorem''' is named after [[Edward Charles Titchmarsh]], | | The title of the writer is Luther. The preferred pastime for him and his kids is to generate and now he is trying to make money with it. Her family lives in Idaho. She works as a financial officer and she will not change it whenever soon.<br><br>My web-site [http://royalcajun.com/UserProfile/tabid/57/userId/17325/Default.aspx http://royalcajun.com/UserProfile/tabid/57/userId/17325/Default.aspx] |
| 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
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| ::<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 | |
| =\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>
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| 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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The title of the writer is Luther. The preferred pastime for him and his kids is to generate and now he is trying to make money with it. Her family lives in Idaho. She works as a financial officer and she will not change it whenever soon.
My web-site http://royalcajun.com/UserProfile/tabid/57/userId/17325/Default.aspx