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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Bibliographic Details
Published in:Mathematics in computer science Vol. 10; no. 3; pp. 365 - 386
Main Authors: Conceição, Ana C., Marreiros, Rui C., Pereira, José C.
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.09.2016
Springer Nature B.V
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:1661-8270
1661-8289
DOI:10.1007/s11786-016-0271-3