Symbolic Computation Applied to the Study of the Kernel of Special Classes of Paired Singular Integral Operators
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 so...
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| Published in: | Mathematics in computer science Vol. 15; no. 1; pp. 63 - 90 |
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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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| AbstractList | 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. |
| Author | Conceição, Ana C. |
| Author_xml | – sequence: 1 givenname: Ana C. orcidid: 0000-0001-7103-3588 surname: Conceição fullname: Conceição, Ana C. email: aconcei@ualg.com organization: Center for Functional Analysis, Linear Structures and Applications (CEAFEL) Departamento de Matemática, Universidade do Algarve |
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| Keywords | Primary 68W30 Factorization algorithms 47G10 Wolfram 47A68 47B35 Kernel of paired singular integral operators Secondary 30E20 Symbolic computation Essentially bounded matrix functions |
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| References | FeldmanIGohbergIKrupnikNAn explicit factorization algorithmIntegral Equ. Oper. Theory2004492149164206037010.1007/s00020-003-1277-1 GohbergIKrupnikNOne-dimensional linear singular integral equationsOper. Theory Adv. Appl.199253199211382080781.47038 CâmaraMCdos SantosAFGeneralised factorization for a class of n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\times n$$\end{document} matrix functions—partial indices and explicit formulasIntegral Equ. Oper. Theory199420219823010.1007/BF01679671 Conceição, A.C.: Factorization of Some Classes of Matrix Functions and its Applications (in Portuguese). Ph.D. thesis, University of Algarve, Faro (2007) RudinWReal and Complex Analysis1987New YorkMcGraw-Hill0925.00005 Conceição, A.C.: Computing the kernel of special classes of paired singular integral operators with Mathematica software. In: Loja, A., Barbosa, J.I., Rodrigues, J.A. (eds.) Proceedings of the 4th International Conference on Numerical and Symbolic Computation: Developments and Applications, Porto - Portugal (2019) Conceição, A.C., Pereira, J.C.: Using wolfram mathematica in spectral theory. In: Loja, A., Barbosa, J.I., Rodrigues, J.A. (eds.) Proceedings of the 3rd International Conference on Numerical and Symbolic Computation: Developments and Applications, Guimarães, Portugal, pp. 295–304 (2017) FeldmanIMarcusAOn some properties of factorization indicesIntegral Equ. Oper Theory1998303326337160868110.1007/BF01195587 VoroninAFA method for determining the partial indices of symmetric matrix functionsSib. Math. J.2011524153281025010.1134/S0037446606010058 Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society: Lecture Note Series, vol. 149. Cambridge University Press, Cambridge (1991) CâmaraMCdos SantosAFCarpentierMExplicit Wiewer–Hopf factorisation and non-linear Riemann–Hilbert problemsProc. R. Soc. Edinb. Sect. (A)20021321457410.1017/S0308210500001529 ConceiçãoACMarreirosRCPereiraJCSymbolic computation applied to the study of the kernel of a singular integral operator with non-Carleman shift and conjugationMath. Comput. Sci.2016103365386354468810.1007/s11786-016-0271-3 ClanceyKGohbergIFactorization of matrix functions and singular integral operatorsOper. Theory Adv. Appl.1981319816577620474.47023 JanashiaGLagvilavaEOn factorization and partial indices of unitary matrix-functions of one classGeorgian Math. J.199745439442146932710.1023/A:1022972315465 ConceiçãoACPereiraJCExploring the spectra of some classes of singular integral operators with symbolic computationMath. Comput. Sci.2016102291309350760610.1007/s11786-016-0264-2 GohbergIKrupnikNOne-dimensional linear singular integral equationsOper. Theory Adv. Appl.199254199211829870781.47038 ConceiçãoACPereiraJCExploring the spectra of some classes of paired singular integral operators: the scalar and matrix casesLib. Math. (new series)20143423533836071342.47006 ConceiçãoACKravchenkoVGAbout explicit factorization of some classes of non-rational matrix functionsMath. Nachr.20072809–1010221034233465710.1002/mana.200510533 EhrhardtTSpeckF-OTransformation techniques towards the factorization of non-rational 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} matrix functionsLinear Algebra Appl.20023531–35390191874910.1016/S0024-3795(02)00288-4 KravchenkoVGLitvinchukGSIntroduction to the Theory of Singular Integral Operators with Shift. Mathematics and its Applications1994DordrechtKluwer0811.47049 PlemeljJRiemannshe Funktionenscharen mit gegebener MonodromiegruppeMonat. Math. Phys.19081921124510.1007/BF01736697 BallJAClanceyKFAn elementary description of partial indices of rational matrix functionsIntegral Equ. Oper. Theory1990133316322104777210.1007/BF01199887 FaddeevLDTkhatayanLAHamiltonian Methods in the Theory of Solitons1987BerlinSpringer10.1007/978-3-540-69969-9 KravchenkoVGMigdal’skiiAIA regularization algorithm for some boundary-value problems of linear conjugationDokl. Math.199552319321 LitvinchukGSSpitkovskiiIMFactorization of measurable matrix functionsOper. Theory Adv. Appl.19872519871015716 ConceiçãoACKravchenkoVGPereiraJCComputing some classes of Cauchy type singular integrals with Mathematica softwareAdv. Comput. Math.2013392273288308251410.1007/s10444-012-9279-7 MikhlinSGPrössdorfSSingular Integral Operators1986BerlinSpringer10.1007/978-3-642-61631-0 AktosunTKlausMvan der MeeCExplicit Wiener–Hopf factorization for certain non-rational matrix functionsIntegral Equ. Oper. Theory199215687990010.1007/BF01203119 ConceiçãoACKravchenkoVGTeixeiraFSSamkoSLebreAdos SantosAFFactorization of some classes of matrix functions and the resolvent of a Hankel operatorFSORP2003 Factorization, Singular Operators and Related Problems, Funchal, Portugal2003DordrechtKluwer10111010.1007/978-94-017-0227-0_8 GohbergIKaashoekMASpitkovskyIMGohbergIManojlovivNdos SantosAFAn overview of matrix factorization theory and operator applicationsFactorization and Integrable Systems, Operator Theory: Advances and Applications2003BaselBirkhäuser110210.1007/978-3-0348-8003-9 CalderónAPCauchy integrals on Lipschitz curves and related operatorsProc. Nat. Acad. Sci. USA19977441324132746656810.1073/pnas.74.4.1324 Conceição, A.C., Kravchenko, V.G., Pereira, J.C.: Rational functions factorization algorithm: a symbolic computation for the scalar and matrix cases. In: Proceedings of the 1st National Conference on Symbolic Computation in Education and Research (CD-ROM), P02, 13 pp. Instituto Superior Técnico, Lisboa, Portugal, April 2–3 (2012) ConceiçãoACMarreirosRCOn the kernel of a singular integral operator with non-Carleman shift and conjugationOper. Matrices201592433456333857510.7153/oam-09-27 LitvinchukGSSolvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Mathematics and its Applications2000DordrechtKluwer0980.45001 PrössdorfSSome Classes of Singular Equations1978AmsterdamNorth-Holland0416.45003 ConceiçãoACKravchenkoVGPereiraJCBallJBolotnikovVRodmanLHeltonJSpitkovskyIFactorization algorithm for some special non-rational matrix functionsTopics in Operator Theory, Operator Theory: Advances and Applications2010BaselBirkhäuser8710910.1007/978-3-0346-0158-0_6 ConceiçãoACKravchenkoVGTeixeiraFSBüttcherAKaashoekMALebreABdos SantosAFSpeckF-OFactorization of matrix funtions and the resolvents of certain operators Singular Integral OperatorsFactorization and Applications—Operator Theory: Advances and Applications2003BaselBirkhäuser911001057.47025 DavisGOpérateurs intégraux singuliers sur certaines courbes du plan complexeAnn. Sci. Ecole Norm. S.198417115718910.24033/asens.1469 Nikol’skiiNKTreatise on the Shift Operator. Spectral Function Theory. Grundlehren der mathematischen Wissenschaften1986BerlinSpringer GarnettJBBounded Analytic Functions. Graduate Texts in Mathematics2007BerlinSpringer AC Conceição (463_CR13) 2010 AC Conceição (463_CR14) 2003 J Plemelj (463_CR37) 1908; 19 GS Litvinchuk (463_CR33) 2000 G Davis (463_CR21) 1984; 17 AC Conceição (463_CR20) 2014; 34 AC Conceição (463_CR11) 2013; 39 463_CR18 VG Kravchenko (463_CR32) 1995; 52 T Aktosun (463_CR2) 1992; 15 VG Kravchenko (463_CR31) 1994 I Gohberg (463_CR28) 1992; 54 I Gohberg (463_CR27) 2003 W Rudin (463_CR39) 1987 SG Mikhlin (463_CR35) 1986 AF Voronin (463_CR40) 2011; 52 MC Câmara (463_CR6) 2002; 132 I Feldman (463_CR25) 1998; 30 S Prössdorf (463_CR38) 1978 LD Faddeev (463_CR23) 1987 JB Garnett (463_CR26) 2007 AC Conceição (463_CR15) 2003 I Feldman (463_CR24) 2004; 49 G Janashia (463_CR30) 1997; 4 AP Calderón (463_CR4) 1997; 74 463_CR1 463_CR12 K Clancey (463_CR7) 1981; 3 T Ehrhardt (463_CR22) 2002; 353 MC Câmara (463_CR5) 1994; 20 I Gohberg (463_CR29) 1992; 53 GS Litvinchuk (463_CR34) 1987; 25 AC Conceição (463_CR10) 2007; 280 AC Conceição (463_CR19) 2016; 10 463_CR9 NK Nikol’skii (463_CR36) 1986 AC Conceição (463_CR17) 2016; 10 JA Ball (463_CR3) 1990; 13 AC Conceição (463_CR16) 2015; 9 463_CR8 |
| References_xml | – reference: ConceiçãoACKravchenkoVGAbout explicit factorization of some classes of non-rational matrix functionsMath. Nachr.20072809–1010221034233465710.1002/mana.200510533 – reference: JanashiaGLagvilavaEOn factorization and partial indices of unitary matrix-functions of one classGeorgian Math. J.199745439442146932710.1023/A:1022972315465 – reference: LitvinchukGSSolvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Mathematics and its Applications2000DordrechtKluwer0980.45001 – reference: ConceiçãoACKravchenkoVGTeixeiraFSBüttcherAKaashoekMALebreABdos SantosAFSpeckF-OFactorization of matrix funtions and the resolvents of certain operators Singular Integral OperatorsFactorization and Applications—Operator Theory: Advances and Applications2003BaselBirkhäuser911001057.47025 – reference: BallJAClanceyKFAn elementary description of partial indices of rational matrix functionsIntegral Equ. Oper. Theory1990133316322104777210.1007/BF01199887 – reference: Conceição, A.C., Pereira, J.C.: Using wolfram mathematica in spectral theory. In: Loja, A., Barbosa, J.I., Rodrigues, J.A. (eds.) Proceedings of the 3rd International Conference on Numerical and Symbolic Computation: Developments and Applications, Guimarães, Portugal, pp. 295–304 (2017) – reference: ConceiçãoACMarreirosRCOn the kernel of a singular integral operator with non-Carleman shift and conjugationOper. Matrices201592433456333857510.7153/oam-09-27 – reference: EhrhardtTSpeckF-OTransformation techniques towards the factorization of non-rational 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} matrix functionsLinear Algebra Appl.20023531–35390191874910.1016/S0024-3795(02)00288-4 – reference: PlemeljJRiemannshe Funktionenscharen mit gegebener MonodromiegruppeMonat. Math. Phys.19081921124510.1007/BF01736697 – reference: ConceiçãoACKravchenkoVGTeixeiraFSSamkoSLebreAdos SantosAFFactorization of some classes of matrix functions and the resolvent of a Hankel operatorFSORP2003 Factorization, Singular Operators and Related Problems, Funchal, Portugal2003DordrechtKluwer10111010.1007/978-94-017-0227-0_8 – reference: ConceiçãoACPereiraJCExploring the spectra of some classes of paired singular integral operators: the scalar and matrix casesLib. Math. (new series)20143423533836071342.47006 – reference: FeldmanIGohbergIKrupnikNAn explicit factorization algorithmIntegral Equ. Oper. Theory2004492149164206037010.1007/s00020-003-1277-1 – reference: Nikol’skiiNKTreatise on the Shift Operator. Spectral Function Theory. Grundlehren der mathematischen Wissenschaften1986BerlinSpringer – reference: FaddeevLDTkhatayanLAHamiltonian Methods in the Theory of Solitons1987BerlinSpringer10.1007/978-3-540-69969-9 – reference: CâmaraMCdos SantosAFCarpentierMExplicit Wiewer–Hopf factorisation and non-linear Riemann–Hilbert problemsProc. R. Soc. Edinb. Sect. (A)20021321457410.1017/S0308210500001529 – reference: Conceição, A.C.: Computing the kernel of special classes of paired singular integral operators with Mathematica software. In: Loja, A., Barbosa, J.I., Rodrigues, J.A. (eds.) Proceedings of the 4th International Conference on Numerical and Symbolic Computation: Developments and Applications, Porto - Portugal (2019) – reference: ConceiçãoACKravchenkoVGPereiraJCComputing some classes of Cauchy type singular integrals with Mathematica softwareAdv. Comput. Math.2013392273288308251410.1007/s10444-012-9279-7 – reference: Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society: Lecture Note Series, vol. 149. Cambridge University Press, Cambridge (1991) – reference: Conceição, A.C., Kravchenko, V.G., Pereira, J.C.: Rational functions factorization algorithm: a symbolic computation for the scalar and matrix cases. In: Proceedings of the 1st National Conference on Symbolic Computation in Education and Research (CD-ROM), P02, 13 pp. Instituto Superior Técnico, Lisboa, Portugal, April 2–3 (2012) – reference: GohbergIKaashoekMASpitkovskyIMGohbergIManojlovivNdos SantosAFAn overview of matrix factorization theory and operator applicationsFactorization and Integrable Systems, Operator Theory: Advances and Applications2003BaselBirkhäuser110210.1007/978-3-0348-8003-9 – reference: MikhlinSGPrössdorfSSingular Integral Operators1986BerlinSpringer10.1007/978-3-642-61631-0 – reference: RudinWReal and Complex Analysis1987New YorkMcGraw-Hill0925.00005 – reference: ConceiçãoACKravchenkoVGPereiraJCBallJBolotnikovVRodmanLHeltonJSpitkovskyIFactorization algorithm for some special non-rational matrix functionsTopics in Operator Theory, Operator Theory: Advances and Applications2010BaselBirkhäuser8710910.1007/978-3-0346-0158-0_6 – reference: GohbergIKrupnikNOne-dimensional linear singular integral equationsOper. Theory Adv. Appl.199254199211829870781.47038 – reference: ConceiçãoACMarreirosRCPereiraJCSymbolic computation applied to the study of the kernel of a singular integral operator with non-Carleman shift and conjugationMath. Comput. Sci.2016103365386354468810.1007/s11786-016-0271-3 – reference: GarnettJBBounded Analytic Functions. Graduate Texts in Mathematics2007BerlinSpringer – reference: DavisGOpérateurs intégraux singuliers sur certaines courbes du plan complexeAnn. Sci. Ecole Norm. S.198417115718910.24033/asens.1469 – reference: PrössdorfSSome Classes of Singular Equations1978AmsterdamNorth-Holland0416.45003 – reference: ConceiçãoACPereiraJCExploring the spectra of some classes of singular integral operators with symbolic computationMath. Comput. Sci.2016102291309350760610.1007/s11786-016-0264-2 – reference: CalderónAPCauchy integrals on Lipschitz curves and related operatorsProc. Nat. Acad. Sci. USA19977441324132746656810.1073/pnas.74.4.1324 – reference: CâmaraMCdos SantosAFGeneralised factorization for a class of n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\times n$$\end{document} matrix functions—partial indices and explicit formulasIntegral Equ. Oper. Theory199420219823010.1007/BF01679671 – reference: Conceição, A.C.: Factorization of Some Classes of Matrix Functions and its Applications (in Portuguese). Ph.D. thesis, University of Algarve, Faro (2007) – reference: FeldmanIMarcusAOn some properties of factorization indicesIntegral Equ. Oper Theory1998303326337160868110.1007/BF01195587 – reference: LitvinchukGSSpitkovskiiIMFactorization of measurable matrix functionsOper. Theory Adv. Appl.19872519871015716 – reference: AktosunTKlausMvan der MeeCExplicit Wiener–Hopf factorization for certain non-rational matrix functionsIntegral Equ. Oper. Theory199215687990010.1007/BF01203119 – reference: KravchenkoVGLitvinchukGSIntroduction to the Theory of Singular Integral Operators with Shift. Mathematics and its Applications1994DordrechtKluwer0811.47049 – reference: GohbergIKrupnikNOne-dimensional linear singular integral equationsOper. Theory Adv. Appl.199253199211382080781.47038 – reference: ClanceyKGohbergIFactorization of matrix functions and singular integral operatorsOper. Theory Adv. Appl.1981319816577620474.47023 – reference: KravchenkoVGMigdal’skiiAIA regularization algorithm for some boundary-value problems of linear conjugationDokl. Math.199552319321 – reference: VoroninAFA method for determining the partial indices of symmetric matrix functionsSib. Math. 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| Title | Symbolic Computation Applied to the Study of the Kernel of Special Classes of Paired Singular Integral Operators |
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