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- Q7524243 subject Q6103418.
- Q7524243 subject Q6960175.
- Q7524243 subject Q7451798.
- Q7524243 abstract "In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The two main singular integral operators, the Hilbert transform and the Cauchy transform, can be defined for any smooth Jordan curve in the complex plane and are related by a simple algebraic formula. The Hilbert transform is an involution and the Cauchy transform an idempotent. The range of the Cauchy transform is the Hardy space of the bounded region enclosed by the Jordan curve. The theory for the original curve can be deduced from that on the unit circle, where, because of rotational symmetry, both operators are classical singular integral operators of convolution type. The Hilbert transform satisfies the jump relations of Plemelj and Sokhotski, which express the original function as the difference between the boundary values of holomorphic functions on the region and its complement. Singular integral operators have been studied on various classes of functions, including Hőlder spaces, Lp spaces and Sobolev spaces. In the case of L2 spaces—the case treated in detail below—other operators associated with the closed curve, such as the Szegő projection onto Hardy space and the Neumann–Poincaré operator, can be expressed in terms of the Cauchy transform and its adjoint.".
- Q7524243 wikiPageWikiLink Q1336492.
- Q7524243 wikiPageWikiLink Q1518047.
- Q7524243 wikiPageWikiLink Q1537963.
- Q7524243 wikiPageWikiLink Q1543149.
- Q7524243 wikiPageWikiLink Q1585233.
- Q7524243 wikiPageWikiLink Q193756.
- Q7524243 wikiPageWikiLink Q1964537.
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- Q7524243 wikiPageWikiLink Q5592049.
- Q7524243 wikiPageWikiLink Q5675112.
- Q7524243 wikiPageWikiLink Q6103418.
- Q7524243 wikiPageWikiLink Q685437.
- Q7524243 wikiPageWikiLink Q6960175.
- Q7524243 wikiPageWikiLink Q7001957.
- Q7524243 wikiPageWikiLink Q7451798.
- Q7524243 wikiPageWikiLink Q7524241.
- Q7524243 wikiPageWikiLink Q7664533.
- Q7524243 wikiPageWikiLink Q769374.
- Q7524243 wikiPageWikiLink Q846862.
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- Q7524243 wikiPageWikiLink Q913764.
- Q7524243 wikiPageWikiLink Q980557.
- Q7524243 type Thing.
- Q7524243 comment "In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The two main singular integral operators, the Hilbert transform and the Cauchy transform, can be defined for any smooth Jordan curve in the complex plane and are related by a simple algebraic formula. The Hilbert transform is an involution and the Cauchy transform an idempotent.".
- Q7524243 label "Singular integral operators on closed curves".
- Q7524243 seeAlso Q1543149.
- Q7524243 seeAlso Q1585233.
- Q7524243 seeAlso Q7664533.