Beam propagation method - Revision history

en>Omnipaedista: the article cannot have two lead section

2015-01-02T08:19:23Z

the article cannot have two lead section

← Older revision		Revision as of 10:19, 2 January 2015
Line 1:		Line 1:
	~~I'm~~ Luther ~~but I by no means truly extended car warranty liked that name~~. ~~Interviewing is what she does but soon she'll be on her own. Her buddies say it's not good~~ for ~~her but what she loves performing~~ is ~~flower arranging~~ and ~~she~~ is ~~[http://direlegion.com/ActivityFeed/MyProfile/tabid/61/UserId/34476/Default.aspx extended car warranty] attempting~~ to make it ~~a profession~~. ~~Arizona is her car [http://www~~.Mass.gov/ocabr/consumer-rights-and-resources/autos/lemon-laws/used-vehicle-warranty-law.html warranty] beginning [https://Autoclubsouth.Aaa.com/Automotive/auto_warranty.aspx?nvbar=Automotive:ExtendedWarranty location] and she will ~~by no means move~~.<br><br>~~Feel free to surf to my blog post:~~ [http://~~checkmates~~.~~co.za~~/~~index.php?do=~~/~~profile-1592~~/~~info~~/ ~~extended auto warranty] ([~~http://~~Royalcajun~~.com/UserProfile/tabid/57/userId/17325/Default.aspx ~~just click the next web site~~])		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]

24.242.9.115: /* See also */

2014-02-28T15:26:51Z

See also

← Older revision		Revision as of 17:26, 28 February 2014
Line 1:		Line 1:
	~~The '''Titchmarsh convolution theorem~~''' is ~~named after [[Edward Charles Titchmarsh]],~~		I'm Luther but I by no means truly extended car warranty liked that name. Interviewing is what she does but soon she'll be on her own. Her buddies say it's not good for her but what she loves performing is flower arranging and she is [http://direlegion.com/ActivityFeed/MyProfile/tabid/61/UserId/34476/Default.aspx extended car warranty] attempting to make it a profession. Arizona is her car [http://www.Mass.gov/ocabr/consumer-rights-and-resources/autos/lemon-laws/used-vehicle-warranty-law.html warranty] beginning [https://Autoclubsouth.Aaa.com/Automotive/auto_warranty.aspx?nvbar=Automotive:ExtendedWarranty location] and she will by no means move.<br><br>Feel free to surf to my blog post: [http://checkmates.co.za/index.php?do=/profile-1592/info/ extended auto warranty] ([http://Royalcajun.com/UserProfile/tabid/57/userId/17325/Default.aspx just click the next web site])
	~~a British mathematician.~~ ~~The theorem describes the properties of the [~~[~~support (mathematics)\|support]] of the [[convolution]] of two functions.~~

	~~== Titchmarsh convolution theorem ==~~
	~~[[Edward Charles Titchmarsh\|E.C. Titchmarsh]] proved the following theorem in 1926:~~

	:~~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>~~
	~~:[[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>~~.

	~~This result, known as the Titchmarsh convolution theorem, could be restated in the following form:~~

	~~: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.~~
	~~: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.~~

	~~This theorem essentially states that the well~~-~~known inclusion~~
	~~:<math>~~
	~~{\rm supp}\,\phi\ast \psi~~
	~~\subset~~
	~~\mathop{\rm supp}\,\phi~~
	~~+\mathop{\rm supp}\,\psi~~
	</~~math>~~
	~~is sharp at the boundary.~~

	~~The higher~~-~~dimensional generalization in terms of the~~
	~~[[convex hull]] of the supports was proved by~~
	~~[[Jacques~~-~~Louis Lions\|J.~~-L. ~~Lions~~]~~] in 1951~~:

	~~: ''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>''~~

	~~Above, <math>\mathop{c~~.~~h.}</math> denotes the [[convex hull~~]~~] of the set~~.
	<~~math~~>~~\mathcal{E}'(\mathbb{R}^n)~~<~~/math~~>
	~~denotes~~
	~~the space of~~ [~~[distribution (mathematics)\|distributions]] with [[compact support]]~~.

	~~The theorem lacks an '''elementary''' proof~~.
	~~The original proof by Titchmarsh~~
	~~is based on the [[Phragmén–Lindelöf principle]],~~
	~~[[Jensen's inequality]],~~
	~~[[Theorem of Carleman]],~~
	~~and~~
	~~[[Bloch's theorem (complex variables)#Valiron's theorem\|Theorem of Valiron]]~~.
	~~More proofs are contained in [Hörmander, Theorem 4.3.3~~] ([~~[harmonic analysis]] style),~~
	~~[Yosida, Chapter VI] ([[real analysis]] style),~~
	~~and [Levin, Lecture 16] ([[complex analysis]] style)~~.

	~~==References==~~

	*{{cite journal
	~~\| author = Titchmarsh, E.C.~~
	~~\| authorlink = Edward Charles Titchmarsh~~
	~~\| title = The zeros of certain integral functions~~
	~~\| journal = [[Proceedings of the London Mathematical Society]]~~
	~~\| volume = 25~~
	~~\| year = 1926~~
	~~\| pages = 283–302~~
	~~\| doi = 10.1112~~/~~plms~~/~~s2-25~~.~~1.283}}~~

	*{{cite journal
	~~\| author = Lions, J.-L.~~
	~~\| title = Supports de produits de composition~~
	~~\| format = I and II~~
	~~\| journal = [[Les Comptes rendus de l'Académie des sciences]]~~
	~~\| volume = 232~~
	~~\| year = 1951~~
	~~\| pages = 1530–1532, 1622–1624}}~~

	*{{cite journal
	~~\| authors = Mikusiński, J. and Świerczkowski, S.~~
	~~\| title = Titchmarsh's theorem on convolution and~~ the ~~theory of Dufresnoy~~
	~~\| journal = [[Prace Matematyczne~~]]
	~~\| volume = 4~~
	~~\| year = 1960~~
	~~\| pages = 59-76}}~~

	*{{cite book
	~~\| author = Yosida, K.~~
	~~\| title = Functional Analysis~~
	~~\| edition = 6th~~
	~~\| series = Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences~~)~~, vol. 123~~
	~~\| publisher = Springer-Verlag~~
	~~\| location = Berlin~~
	~~\| year = 1980}}~~

	*{{cite book
	~~\| authorlink = Lars Hörmander~~
	~~\| author = Hörmander, L.~~
	~~\| title = The Analysis of Linear Partial Differential Operators, I~~
	~~\| edition = 2nd~~
	~~\| series = Springer Study Edition~~
	~~\| publisher = Springer-Verlag~~
	~~\| location = Berlin~~
	~~\| year = 1990}}~~

	*{{cite book
	~~\| author = Levin, B. Ya.~~
	~~\| title = Lectures on Entire Functions~~
	~~\| series = Translations of Mathematical Monographs, vol. 150~~
	~~\| publisher = American Mathematical Society~~
	~~\| location = Providence, RI~~
	~~\| year = 1996}}~~



	~~[[Category:Theorems in harmonic analysis]]~~
	~~[[Category:Theorems in complex analysis]]~~
	~~[[Category:Theorems in real analysis]]~~

216.209.84.26 at 13:40, 8 November 2013

2013-11-08T13:40:48Z

← Older revision		Revision as of 15:40, 8 November 2013
Line 1:		Line 1:
	The ~~title~~ of the ~~author is Garland~~. ~~Meter studying is where my main income comes from but quickly I'll be on my personal~~. ~~What she loves doing is taking part~~ in ~~croquet~~ and ~~she~~ is ~~trying to make it a profession~~. ~~Arizona has always been my living place but my wife wants us to transfer~~.<br><br>~~Look into my blog~~: ~~[http~~://~~Mwarrenassociates~~.~~com~~/~~UserProfile~~/~~tabid~~/~~253~~/~~userId/228462/Default~~.~~aspx cars warranty~~]		The '''Titchmarsh convolution theorem''' is named after [[Edward Charles Titchmarsh]],
			a British mathematician. The theorem describes the properties of the [[support (mathematics)\|support]] of the [[convolution]] of two functions.

			== Titchmarsh convolution theorem ==
			[[Edward Charles Titchmarsh\|E.C. Titchmarsh]] proved the following theorem in 1926:

			: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>
			:[[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>.

			This result, known as the Titchmarsh convolution theorem, could be restated in the following form:

			: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.
			: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.

			This theorem essentially states that the well-known inclusion
			:<math>
			{\rm supp}\,\phi\ast \psi
			\subset
			\mathop{\rm supp}\,\phi
			+\mathop{\rm supp}\,\psi
			</math>
			is sharp at the boundary.

			The higher-dimensional generalization in terms of the
			[[convex hull]] of the supports was proved by
			[[Jacques-Louis Lions\|J.-L. Lions]] in 1951:

			: ''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>''

			Above, <math>\mathop{c.h.}</math> denotes the [[convex hull]] of the set.
			<math>\mathcal{E}'(\mathbb{R}^n)</math>
			denotes
			the space of [[distribution (mathematics)\|distributions]] with [[compact support]].

			The theorem lacks an '''elementary''' proof.
			The original proof by Titchmarsh
			is based on the [[Phragmén–Lindelöf principle]],
			[[Jensen's inequality]],
			[[Theorem of Carleman]],
			and
			[[Bloch's theorem (complex variables)#Valiron's theorem\|Theorem of Valiron]].
			More proofs are contained in [Hörmander, Theorem 4.3.3] ([[harmonic analysis]] style),
			[Yosida, Chapter VI] ([[real analysis]] style),
			and [Levin, Lecture 16] ([[complex analysis]] style).

			==References==

			*{{cite journal
			\| author = Titchmarsh, E.C.
			\| authorlink = Edward Charles Titchmarsh
			\| title = The zeros of certain integral functions
			\| journal = [[Proceedings of the London Mathematical Society]]
			\| volume = 25
			\| year = 1926
			\| pages = 283–302
			\| doi = 10.1112/plms/s2-25.1.283}}

			*{{cite journal
			\| author = Lions, J.-L.
			\| title = Supports de produits de composition
			\| format = I and II
			\| journal = [[Les Comptes rendus de l'Académie des sciences]]
			\| volume = 232
			\| year = 1951
			\| pages = 1530–1532, 1622–1624}}

			*{{cite journal
			\| authors = Mikusiński, J. and Świerczkowski, S.
			\| title = Titchmarsh's theorem on convolution and the theory of Dufresnoy
			\| journal = [[Prace Matematyczne]]
			\| volume = 4
			\| year = 1960
			\| pages = 59-76}}

			*{{cite book
			\| author = Yosida, K.
			\| title = Functional Analysis
			\| edition = 6th
			\| series = Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123
			\| publisher = Springer-Verlag
			\| location = Berlin
			\| year = 1980}}

			*{{cite book
			\| authorlink = Lars Hörmander
			\| author = Hörmander, L.
			\| title = The Analysis of Linear Partial Differential Operators, I
			\| edition = 2nd
			\| series = Springer Study Edition
			\| publisher = Springer-Verlag
			\| location = Berlin
			\| year = 1990}}

			*{{cite book
			\| author = Levin, B. Ya.
			\| title = Lectures on Entire Functions
			\| series = Translations of Mathematical Monographs, vol. 150
			\| publisher = American Mathematical Society
			\| location = Providence, RI
			\| year = 1996}}



			[[Category:Theorems in harmonic analysis]]
			[[Category:Theorems in complex analysis]]
			[[Category:Theorems in real analysis]]

en>Vbrulis: /* Limitations of BPM */

2012-06-26T15:19:21Z

Limitations of BPM

New page

The title of the author is Garland. Meter studying is where my main income comes from but quickly I'll be on my personal. What she loves doing is taking part in croquet and she is trying to make it a profession. Arizona has always been my living place but my wife wants us to transfer.<br><br>Look into my blog: [http://Mwarrenassociates.com/UserProfile/tabid/253/userId/228462/Default.aspx cars warranty]