About: Singular integral operators on closed curves

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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. In the special case of Fourier series for the unit circle, the operators become the classical Cauchy transform, the orthogonal projection onto Hardy space, and the Hilbert transform a real orthogonal linear complex structure. In general the Cauchy transform is a non-self-adjoint idempotent and the Hilbert transform a non-orthogonal complex structure. The range of the Cauchy transform is the Hardy space of the bounded region enclosed by the Jordan cu

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rdf:type	Thing
rdfs:label	Singular integral operators on closed curves (en)
rdfs: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. In the special case of Fourier series for the unit circle, the operators become the classical Cauchy transform, the orthogonal projection onto Hardy space, and the Hilbert transform a real orthogonal linear complex structure. In general the Cauchy transform is a non-self-adjoint idempotent and the Hilbert transform a non-orthogonal complex structure. The range of the Cauchy transform is the Hardy space of the bounded region enclosed by the Jordan cu (en)
rdfs:seeAlso	Hardy space Sokhotski–Plemelj theorem Szegő kernel
dcterms:subject	Operator theory Harmonic analysis Complex analysis
Wikipage page ID	36924582 (xsd:integer)
Wikipage revision ID	1075057390 (xsd:integer)
Link from a Wikipage to another Wikipage	Mergelyan's theorem Bergman space Operator theory Complex analysis Analysis Harmonic analysis Mathematics Operator norm Green's theorem Singular integral operators of convolution type Hardy space Harmonic analysis Cauchy's integral formula Cauchy principal value Hartogs–Rosenthal theorem Linear complex structure Neumann–Poincaré operator American Mathematical Society Curvature Fourier series Carathéodory kernel theorem Graduate Studies in Mathematics Hilbert–Schmidt operator Hilbert transform Jean Frédéric Frenet Sokhotski–Plemelj theorem Complex analysis Hölder continuous Idempotent Orthogonal projection Strong operator topology Szegő kernel Joseph Leonard Walsh Residue calculus Singular integral operators
sameAs	Singular integral operators on closed curves Singular integral operators on closed curves Singular integral operators on closed curves
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Harvtxt dbt:Main dbt:Reflist dbt:See_also
has 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. In the special case of Fourier series for the unit circle, the operators become the classical Cauchy transform, the orthogonal projection onto Hardy space, and the Hilbert transform a real orthogonal linear complex structure. In general the Cauchy transform is a non-self-adjoint idempotent and the Hilbert transform a non-orthogonal complex structure. 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 of 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. (en)
prov:wasDerivedFrom	wikipedia-en:Singular_integral_operators_on_closed_curves?oldid=1075057390&ns=0
page length (characters) of wiki page	29271 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Singular_integral_operators_on_closed_curves
is rdfs:seeAlso of	Neumann–Poincaré operator Grunsky matrix
is Link from a Wikipage to another Wikipage of	Singular integral Neumann–Poincaré operator Sokhotski–Plemelj theorem Cauchy singular operator
is Wikipage redirect of	Cauchy singular operator
is foaf:primaryTopic of	wikipedia-en:Singular_integral_operators_on_closed_curves

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