Symbolic Computation Applied to the Study of the Kernel of a Singular Integral Operator with Non-Carleman Shift and Conjugation

On the Hilbert space L ~ 2 ( T ) the singular integral operator with non-Carleman shift and conjugation K = P + + ( a I + A C ) P - is considered, where P ± are the Cauchy projectors, A = ∑ j = 0 m a j U j , a , a j , j = 1 , m ¯ , are continuous functions on the unit circle T , U is the shift opera...

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Vydáno v:	Mathematics in computer science Ročník 10; číslo 3; s. 365 - 386
Hlavní autoři:	Conceição, Ana C., Marreiros, Rui C., Pereira, José C.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Cham Springer International Publishing 01.09.2016 Springer Nature B.V
Témata:	Algorithms Computation Computer algebra Computer Science Conjugation Hilbert space Mathematics Mathematics and Statistics Operators (mathematics) Projectors Non-Carleman shift 68W30 Factorization algorithms Wolfram 47A68 Secondary 45P05 Conjugation Kernel dimension Symbolic computation Primary 47G10 Singular integral operators
ISSN:	1661-8270, 1661-8289
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Abstract	On the Hilbert space L ~ 2 ( T ) the singular integral operator with non-Carleman shift and conjugation K = P + + ( a I + A C ) P - is considered, where P ± are the Cauchy projectors, A = ∑ j = 0 m a j U j , a , a j , j = 1 , m ¯ , are continuous functions on the unit circle T , U is the shift operator and C is the operator of complex conjugation. We show how the symbolic computation capabilities of the computer algebra system Mathematica can be used to explore the dimension of the kernel of the operator K . The analytical algorithm [ADimKer-NonCarleman] is presented; several nontrivial examples are given.
AbstractList	On the Hilbert space L ~ 2 ( T ) the singular integral operator with non-Carleman shift and conjugation K = P + + ( a I + A C ) P - is considered, where P ± are the Cauchy projectors, A = ∑ j = 0 m a j U j , a , a j , j = 1 , m ¯ , are continuous functions on the unit circle T , U is the shift operator and C is the operator of complex conjugation. We show how the symbolic computation capabilities of the computer algebra system Mathematica can be used to explore the dimension of the kernel of the operator K. The analytical algorithm [ADimKer-NonCarleman] is presented; several nontrivial examples are given. On the Hilbert space L ~ 2 ( T ) the singular integral operator with non-Carleman shift and conjugation K = P + + ( a I + A C ) P - is considered, where P ± are the Cauchy projectors, A = ∑ j = 0 m a j U j , a , a j , j = 1 , m ¯ , are continuous functions on the unit circle T , U is the shift operator and C is the operator of complex conjugation. We show how the symbolic computation capabilities of the computer algebra system Mathematica can be used to explore the dimension of the kernel of the operator K . The analytical algorithm [ADimKer-NonCarleman] is presented; several nontrivial examples are given.
Author	Marreiros, Rui C. Conceição, Ana C. Pereira, José C.
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Snippet	On the Hilbert space L ~ 2 ( T ) the singular integral operator with non-Carleman shift and conjugation K = P + + ( a I + A C ) P - is considered, where P ±... On the Hilbert space L ~ 2 ( T ) the singular integral operator with non-Carleman shift and conjugation K = P + + ( a I + A C ) P - is considered, where P ±...
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SubjectTerms	Algorithms Computation Computer algebra Computer Science Conjugation Hilbert space Mathematics Mathematics and Statistics Operators (mathematics) Projectors
Title	Symbolic Computation Applied to the Study of the Kernel of a Singular Integral Operator with Non-Carleman Shift and Conjugation
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