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Mathematics in Computer Science

Erschienen in:

01.04.2020

Symbolic Computation Applied to the Study of the Kernel of Special Classes of Paired Singular Integral Operators

verfasst von: Ana C. Conceição

Erschienen in: Mathematics in Computer Science | Ausgabe 1/2021

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Abstract

Operator theory has many applications in several main scientific research areas (structural mechanics, aeronautics, quantum mechanics, ecology, probability theory, electrical engineering, among others) and the importance of its study is globally acknowledged. On the study of the operator’s kernel some progress has been achieved for some specific classes of singular integral operators whose properties allow the use of particular strategies. However, the existing algorithms allow, in general, to study the dimension of the kernel of some classes of singular integral operators but are not designed to be implemented on a computer. The main goal of this paper is to show how the symbolic and numeric capabilities of a computer algebra system can be used to study the kernel of special classes of paired singular integral operators with essentially bounded coefficients defined on the unit circle. It is described how some factorization algorithms can be used to compute the dimension of the kernel of special classes of singular integral operators. The analytical algorithms [ADimKerPaired-Scalar], [AKerPaired-Scalar], and [ADimKerPaired-Matrix] are presented. The design of these new algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. For the essentially bounded hermitian coefficients case, there exist some relations with Hankel operators. The paper contains some interesting and nontrivial examples obtained with the use of a computer algebra system.

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Wolfram Mathematica is a symbolic mathematical computation program used in many scientific, engineering, and computing fields. It was conceived by Stephen Wolfram and is developed by Wolfram Research.

Although many of the results presented in this paper can be generalized [4, 21] to the space \(L_p(\Gamma )\), where \(\Gamma \) is a closed Carleson curve, we decided to state them only for

https://static-content.springer.com/image/art%3A10.1007%2Fs11786-020-00463-3/MediaObjects/11786_2020_463_IEq4_HTML.gif

due to the use of symbolic computation for the construction of nontrivial examples.

The [SInt] algorithm described in [11] computes (1) when the essentially bounded function \(\varphi \) can be represented as \(\varphi (t)=r(t)[x_+(t)+y_-(t)]\), where

https://static-content.springer.com/image/art%3A10.1007%2Fs11786-020-00463-3/MediaObjects/11786_2020_463_IEq19_HTML.gif

and r is a rational function without poles on

https://static-content.springer.com/image/art%3A10.1007%2Fs11786-020-00463-3/MediaObjects/11786_2020_463_IEq20_HTML.gif

The corresponding source code of [SInt] is available in the online version of [11].

Some of our analytic algorithms [11, 13, 17, 19] provide us extra information about the class of inner functions.

An analogous result can also be stated for the case when

https://static-content.springer.com/image/art%3A10.1007%2Fs11786-020-00463-3/MediaObjects/11786_2020_463_IEq82_HTML.gif

changing \(\left( I\pm P_{\mp }a^{-1}bP_{\pm }\right) a^{\mp 1}I\) by \((I\pm P_{\pm }ab^{-1}P_{\mp })b^{\mp 1}I\).

A similar result can be obtained for the case when

https://static-content.springer.com/image/art%3A10.1007%2Fs11786-020-00463-3/MediaObjects/11786_2020_463_IEq106_HTML.gif

and the matrix function \(b^{-1}a\) admits a right generalized factorization in

https://static-content.springer.com/image/art%3A10.1007%2Fs11786-020-00463-3/MediaObjects/11786_2020_463_IEq108_HTML.gif

. In this case, the dimension of the kernel of the paired singular integral operator \(T_{\{a,b\}}\) corresponds to the sum of the modulus of the right negative partial indices of \(b^{-1}a\).

Similar results can be obtained for the case when

https://static-content.springer.com/image/art%3A10.1007%2Fs11786-020-00463-3/MediaObjects/11786_2020_463_IEq119_HTML.gif

and the matrix function \(ab^{-1}\) admits a right generalized factorization in

https://static-content.springer.com/image/art%3A10.1007%2Fs11786-020-00463-3/MediaObjects/11786_2020_463_IEq121_HTML.gif

. In this case, the dimension of the kernel of the paired singular integral operator corresponds to the sum of the modulus of the right negative partial indices of \(ab^{-1}\).

A similar result can be obtained for the case when

https://static-content.springer.com/image/art%3A10.1007%2Fs11786-020-00463-3/MediaObjects/11786_2020_463_IEq131_HTML.gif

and the matrix function \(b^{-1}a\) admits a right generalized factorization in

https://static-content.springer.com/image/art%3A10.1007%2Fs11786-020-00463-3/MediaObjects/11786_2020_463_IEq133_HTML.gif

[28] contains formulas to compute the kernels of \(T_{\{a,b\}}\) and \({\widetilde{T}}_{\{a,b\}}\) using the factors of a factorization of the function \(ab^{-1}\).

Similar results can be obtained for the case when the matrix function \(ba^{-1}\) admits a left generalized factorization in

https://static-content.springer.com/image/art%3A10.1007%2Fs11786-020-00463-3/MediaObjects/11786_2020_463_IEq190_HTML.gif

The pretty-print functionality allows to write on the computer screen scientific formulas in the traditional format, as if one was using pencil and paper.

A similar algorithm can be described considering a factorization of the function \(ab^{-1}\).

https://static-content.springer.com/image/art%3A10.1007%2Fs11786-020-00463-3/MediaObjects/11786_2020_463_IEq316_HTML.gif

, then a similar algorithm can be described using the matrix functions \(b^{-1}a\) and \(ab^{-1}\).

The same idea can be applied when \(b^{-1}a\) belongs to class (18) and admits a right generalized factorization in

https://static-content.springer.com/image/art%3A10.1007%2Fs11786-020-00463-3/MediaObjects/11786_2020_463_IEq385_HTML.gif

The same idea can be applied when \(ab^{-1}\) belongs to class (18) and admits a right generalized factorization in

https://static-content.springer.com/image/art%3A10.1007%2Fs11786-020-00463-3/MediaObjects/11786_2020_463_IEq391_HTML.gif

Considering a left generalized factorization (5) of the matrix function \(ba^{-1}\) or a right generalized factorization (5) of the matrix function \(ab^{-1}\).

This condition is provided explicitly in the output of the [AFact] algorithm. It arises from the construction of an homogeneous linear system which we know to be uniquely solvable when \(-\gamma \in \sigma \left( P_- {\overline{\varphi }}P_+ \varphi P_-\right) \).

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Titel: Symbolic Computation Applied to the Study of the Kernel of Special Classes of Paired Singular Integral Operators
verfasst von: Ana C. Conceição
Publikationsdatum: 01.04.2020
Verlag: Springer International Publishing
Erschienen in: Mathematics in Computer Science / Ausgabe 1/2021
Print ISSN: 1661-8270
Elektronische ISSN: 1661-8289
DOI: https://doi.org/10.1007/s11786-020-00463-3

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