en>Illia Connell: /* References */standardize journal name, replaced: Proceedings of the National Academy of the USA → Proceedings of the National Academy of Sciences of the United States of America using AWB

2013-05-26T23:47:51Z

References: standardize journal name, replaced: Proceedings of the National Academy of the USA → Proceedings of the National Academy of Sciences of the United States of America using AWB

← Older revision		Revision as of 01:47, 27 May 2013
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	~~The author~~ is ~~called Irwin~~. ~~What I love performing~~ is ~~playing baseball but I haven~~'~~t made~~ a ~~dime~~ with it. ~~Puerto Rico~~ is ~~exactly where he~~'s ~~always been living but she requirements to transfer because~~ of ~~her family members. Hiring is my profession.~~<br><br>~~My weblog~~: [~~http~~:~~//tdc.flyingv.me/content/cardiovascular~~-~~illnesses-their-causes-signs-and-symptoms at home std test]~~		{{Other uses2\|Disc}}
			In [[function theory]], the '''disk algebra''' ''A''('''D''') (also spelled '''disc algebra''') is the set of [[holomorphic function]]s

			:''f'' : '''D''' → '''C''',

			where '''D''' is the [[open unit disk]] in the [[complex plane]] '''C''', ''f'' extends to a continuous function on the [[closure (topology)\|closure]] of '''D'''. That is,
			:<math>A(\mathbf{D}) = H^\infty(\mathbf{D})\cap C(\overline{\mathbf{D}}),</math>
			where <math>H^\infty(\mathbf{D})</math> denotes the Banach space of bounded analytic functions on the unit disc '''D''' (i.e. a [[Hardy space]]).
			When endowed with the pointwise addition,
			(f+g)(z)=f(z)+g(z),
			and pointwise multiplication,
			:(fg)(z)=f(z)g(z),

			this set becomes an [[algebra over a field\|algebra]] over '''C''', since if ''&fnof;'' and ''g'' belong to the disk algebra then so do ''ƒ'' + ''g'' and ''ƒg''.

			Given the [[uniform norm]],
			:<math>\\|f\\| = \sup\{\|f(z)\|\mid z\in \mathbf{D}\}=\max\{ \|f(z)\|\mid z\in \overline{\mathbf{D}}\},</math>
			by construction it becomes a [[uniform algebra]] and a commutative [[Banach algebra]].

			By construction the disc algebra is a closed subalgebra of the [[Hardy space]] [[H infinity\|''H''<sup>∞</sup>]]. In contrast to the stronger requirement that a continuous extension to the circle exists, it is a lemma of [[Fatou's lemma\|Fatou]] that a general element of [[H infinity\|''H''<sup>∞</sup>]] can be radially extended to the circle [[almost everywhere]].

			[[Category:Functional analysis]]
			[[Category:Complex analysis]]
			[[Category:Banach algebras]]


			{{mathanalysis-stub}}

en>Headbomb: Various citation cleanup, typos fixed: , → , using AWB

2011-09-03T23:25:24Z

Various citation cleanup, typos fixed: , → , using AWB

New page

The author is called Irwin. What I love performing is playing baseball but I haven't made a dime with it. Puerto Rico is exactly where he's always been living but she requirements to transfer because of her family members. Hiring is my profession.<br><br>My weblog: [http://tdc.flyingv.me/content/cardiovascular-illnesses-their-causes-signs-and-symptoms at home std test]

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en>Illia Connell: /* References */standardize journal name, replaced: Proceedings of the National Academy of the USA → Proceedings of the National Academy of Sciences of the United States of America using AWB

en>Headbomb: Various citation cleanup, typos fixed: , → , using AWB