Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2021, Cilt: 4 Sayı: 2, 186 - 214, 01.06.2021
https://doi.org/10.33205/cma.868272

Öz

Kaynakça

  • I. T. Abu-Jeib: Algorithms for Centrosymmetric and Skew-Centrosymmetric Matrices, Missouri J. Math. Sci., 18 (1) (2006), 1-8.
  • F. Altomare, M. Campiti: Korovkin-type approximation theory and its applications, De Gruyter Studies in Mathematics, 17, Walter de Gruyter & C., Berlin (1994).
  • F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Rasa: Markov operators, positive semigroups and approximation processes, De Gruyter Studies in Mathematics 61, De Gruyter, Berlin (2014).
  • P. N. Agrawal, H. S. Kasana: On iterative combinations of Bernstein polynomials, Demonstr. Math. 17, (1984) 777–783.
  • U. Amato, B. Della Vecchia: Bridging Bernstein and Lagrange polynomials, Math. Commun., 20 (2) (2015), 151–160.
  • K. E. Atkinson: The Numerical Solution of Integral Equations of the second kind, Cambridge Monographs on Applied and Computational Mathematics, 4. Cambridge University Press, Cambridge (1997).
  • H. Brass, J. W. Fischer: Error bounds for Romberg quadrature, Numer. Math., 82 (3) (1999), 389-408.
  • M. Campiti: Convergence of Iterated Boolean-type Sums and Their Iterates, Numer. Funct. Anal. Optim., 39 (10) (2018), 1054-1063.
  • M. R. Capobianco, G. Mastroianni and M. G. Russo: Pointwise and uniform approximation of the finite Hilbert transform, "Approximation and Optimization, Proceedings of International Conference on Approximation and Optimization (ICAOR) Cluj-Napoca, July 29-August 1, 1996, (Eds. Stancu D. Coman G., Breckner W.W., Blaga P.) 1 (1997), 45-66.
  • S. Cooper, S. Waldron: The eigenstructure of the Bernstein operator, J. Approx. Theory, 105 (1) (2000), 133-165.
  • P. J. Davis, P. Rabinowitz: Methods of numerical integration, Computer Science and Applied Mathematics, Academic Press Inc., Orlando, FL (1984).
  • M. C. De Bonis, G. Mastroianni: Projection methods and condition numbers in uniform norm for Fredholm and Cauchy singular integral equations, SIAM J. Numer. Anal., 44 (4) (2006), 1–24.
  • Z. Ditzian, V. Totik: Moduli of smoothness, Springer Series in Computational Mathematics 9, Springer-Verlag, New York (1987).
  • Z. Ditzian, V. Totik: Remarks on Besov spaces and best polynomial approximation, Proc. Amer. Math. Soc., 104 (4) (1988).
  • B. R. Draganov: Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their Boolean sums, J. Approx. Theory, 200 (2015), 92-135.
  • G. Farin: Curves and surfaces for computer aided geometric design. A practical guide, Third edition. Computer Science and Scientific Computing. Academic Press, Inc., Boston, MA (1993)
  • G. Felbecker: Linearkombinationen von iterierten Bernsteinoperatoren, Manuscripta Math., 29 (2-4) (1979), 229-246
  • F. Filbir, D. Occorsio and W. Themistoclakis: Approximation of Finite Hilbert and Hadamard transforms by using equally spaced nodes, Mathematics, 8 (4) (2020), Article number 542.
  • H. H. Gonska, X. L. Zhou: Approximation theorems for the iterated Boolean sums of Bernstein operators, J. Comput. Appl. Math., 53 (1994) 21-31.
  • P. Junghanns, U. Luther: Cauchy singular integral equations in spaces of continuous functions and methods for their numerical solution, ROLLS Symposium (Leipzig, 1996). J. Comput. Appl. Math., 77 (1-2) (1997), 201–237.
  • A. I. Kalandiya: Mathematical Methods of Two-Dimensional Elasticity, Publ., Nauka Moscow (1973).
  • F. King: Hilbert Transforms, I & II. Cambridge University Press, Cambridge (2009).
  • G. G. Lorentz: Bernstein polynomials, Second edition. Chelsea Publishing Co., New York (1986)
  • G. Mastroianni, M. R. Occorsio: Una generalizzazione dell’operatore di Bernstein, Rend. dell’Accad. di Scienze Fis. e Mat. Napoli (Serie IV), 44 (1977), 151–169
  • G. Mastroianni, M. R. Occorsio: Alcuni algoritmi per il calcolo numerico di integrali a valor principale secondo Cauchy, (available in Italian only) Rapporto Tecnico I.A.M. 3/84.
  • G. Mastroianni, M.R. Occorsio: An algorithm for the numerical evaluation of a Cauchy principal value integral, Ricerche Mat., 33 (1) (1984), 3-18.
  • G. Mastroianni, M. G. Russo and W. Themistoclakis: Numerical Methods for Cauchy Singular Integral Equations in Spaces of Weighted Continuous Functions, Operator Theory Advances and Applications, 160, Birkhäuser Verlag Basel, Switzerland (2005), 311-336.
  • G. Mastroianni, M. G. Russo and W. Themistoclakis: The boundedness of the Cauchy singular integral operator in weighted Besov type spaces with uniform norms, Integr. Equ. Oper. Theory, 42, Birkhäuser Verlag Basel (2002), 57-89.
  • G. Mastroianni, W. Themistoclakis: A numerical method for the generalized airfoil equation based on the de la Vallée Poussin interpolation, J. Comput. Appl. Math., 180 (2005), 71-105.
  • C. Micchelli: The saturation class and iterates of Bernstein polynomials, J. Approx. Th., 8 (1973), 1–18
  • G. Monegato: Numerical evaluation of hypersingular integrals, J. Comp. Appl. Math., 50 (1994), 9-31.
  • D. Occorsio, M. G. Russo: A Nyström method for Fredholm integral equations based on equally spaced knots, Filomat, 28 (1) (2014), 49-63.
  • D. Occorsio, M. G. Russo: Bivariate Generalized Bernstein Operators and their application to Fredholm Integral Equations, Publ. Inst. Math. N. S., 100 (114) (2016), 141-162.
  • D. Occorsio, A. C. Simoncelli: How to go from Bézier to Lagrange curves by means of generalized Bézier curves, Facta Univ. Ser. Math. Inform. (Niš), 11 (1996), 101-111.
  • D. Occorsio: Some new properties of Generalized Bernstein polynomials, Stud. Univ. Babe¸s-Bolyai Math, 56 (3) (2011), 147-160.
  • I. Rasa: Iterated Boolean sums of Bernstein and related operators, Rev. Anal. Numér. Théor. Approx, 35 (2006), 111-115.
  • S. Weiwei, W. Jiming: Interpolatory quadrature rules for Hadamard finite-part integrals and their superconvergence, IMA J. Numer. Anal., 28 (2008), 580-597.
  • G. Tachev: From Bernstein polynomials to Lagrange interpolation, Proceedings of 2nd International Conference on Modelling and Development of Intelligent Systems (MDIS 2011), (2011), 192-197.

Some numerical applications of generalized Bernstein operators

Yıl 2021, Cilt: 4 Sayı: 2, 186 - 214, 01.06.2021
https://doi.org/10.33205/cma.868272

Öz

In this paper some recent applications of the so-called Generalized Bernstein polynomials are collected.
This polynomial sequence is constructed by means of the samples of a continuous function f on equispaced points of
[0; 1] and depends on an additional parameter which yields the remarkable property of improving the rate of convergence
to the function f, according with the smoothness of f. This means that the sequence does not suffer of
the saturation phenomena occurring by using the classical Bernstein polynomials or arising in piecewise polynomial
approximation. The applications considered here deal with the numerical integration and the simultaneous approximation.
Quadrature rules on equidistant nodes of [0; 1] are studied for the numerical computation of ordinary integrals
in one or two dimensions, and usefully employed in Nyström methods for solving Fredholm integral equations. Moreover,
the simultaneous approximation of the Hilbert transform and its derivative (the Hadamard transform) is illustrated.
For all the applications, some numerical details are given in addition to the error estimates, and the proposed
approximation methods have been implemented providing numerical tests which confirm the theoretical estimates.
Some open problems are also introduced.

Kaynakça

  • I. T. Abu-Jeib: Algorithms for Centrosymmetric and Skew-Centrosymmetric Matrices, Missouri J. Math. Sci., 18 (1) (2006), 1-8.
  • F. Altomare, M. Campiti: Korovkin-type approximation theory and its applications, De Gruyter Studies in Mathematics, 17, Walter de Gruyter & C., Berlin (1994).
  • F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Rasa: Markov operators, positive semigroups and approximation processes, De Gruyter Studies in Mathematics 61, De Gruyter, Berlin (2014).
  • P. N. Agrawal, H. S. Kasana: On iterative combinations of Bernstein polynomials, Demonstr. Math. 17, (1984) 777–783.
  • U. Amato, B. Della Vecchia: Bridging Bernstein and Lagrange polynomials, Math. Commun., 20 (2) (2015), 151–160.
  • K. E. Atkinson: The Numerical Solution of Integral Equations of the second kind, Cambridge Monographs on Applied and Computational Mathematics, 4. Cambridge University Press, Cambridge (1997).
  • H. Brass, J. W. Fischer: Error bounds for Romberg quadrature, Numer. Math., 82 (3) (1999), 389-408.
  • M. Campiti: Convergence of Iterated Boolean-type Sums and Their Iterates, Numer. Funct. Anal. Optim., 39 (10) (2018), 1054-1063.
  • M. R. Capobianco, G. Mastroianni and M. G. Russo: Pointwise and uniform approximation of the finite Hilbert transform, "Approximation and Optimization, Proceedings of International Conference on Approximation and Optimization (ICAOR) Cluj-Napoca, July 29-August 1, 1996, (Eds. Stancu D. Coman G., Breckner W.W., Blaga P.) 1 (1997), 45-66.
  • S. Cooper, S. Waldron: The eigenstructure of the Bernstein operator, J. Approx. Theory, 105 (1) (2000), 133-165.
  • P. J. Davis, P. Rabinowitz: Methods of numerical integration, Computer Science and Applied Mathematics, Academic Press Inc., Orlando, FL (1984).
  • M. C. De Bonis, G. Mastroianni: Projection methods and condition numbers in uniform norm for Fredholm and Cauchy singular integral equations, SIAM J. Numer. Anal., 44 (4) (2006), 1–24.
  • Z. Ditzian, V. Totik: Moduli of smoothness, Springer Series in Computational Mathematics 9, Springer-Verlag, New York (1987).
  • Z. Ditzian, V. Totik: Remarks on Besov spaces and best polynomial approximation, Proc. Amer. Math. Soc., 104 (4) (1988).
  • B. R. Draganov: Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their Boolean sums, J. Approx. Theory, 200 (2015), 92-135.
  • G. Farin: Curves and surfaces for computer aided geometric design. A practical guide, Third edition. Computer Science and Scientific Computing. Academic Press, Inc., Boston, MA (1993)
  • G. Felbecker: Linearkombinationen von iterierten Bernsteinoperatoren, Manuscripta Math., 29 (2-4) (1979), 229-246
  • F. Filbir, D. Occorsio and W. Themistoclakis: Approximation of Finite Hilbert and Hadamard transforms by using equally spaced nodes, Mathematics, 8 (4) (2020), Article number 542.
  • H. H. Gonska, X. L. Zhou: Approximation theorems for the iterated Boolean sums of Bernstein operators, J. Comput. Appl. Math., 53 (1994) 21-31.
  • P. Junghanns, U. Luther: Cauchy singular integral equations in spaces of continuous functions and methods for their numerical solution, ROLLS Symposium (Leipzig, 1996). J. Comput. Appl. Math., 77 (1-2) (1997), 201–237.
  • A. I. Kalandiya: Mathematical Methods of Two-Dimensional Elasticity, Publ., Nauka Moscow (1973).
  • F. King: Hilbert Transforms, I & II. Cambridge University Press, Cambridge (2009).
  • G. G. Lorentz: Bernstein polynomials, Second edition. Chelsea Publishing Co., New York (1986)
  • G. Mastroianni, M. R. Occorsio: Una generalizzazione dell’operatore di Bernstein, Rend. dell’Accad. di Scienze Fis. e Mat. Napoli (Serie IV), 44 (1977), 151–169
  • G. Mastroianni, M. R. Occorsio: Alcuni algoritmi per il calcolo numerico di integrali a valor principale secondo Cauchy, (available in Italian only) Rapporto Tecnico I.A.M. 3/84.
  • G. Mastroianni, M.R. Occorsio: An algorithm for the numerical evaluation of a Cauchy principal value integral, Ricerche Mat., 33 (1) (1984), 3-18.
  • G. Mastroianni, M. G. Russo and W. Themistoclakis: Numerical Methods for Cauchy Singular Integral Equations in Spaces of Weighted Continuous Functions, Operator Theory Advances and Applications, 160, Birkhäuser Verlag Basel, Switzerland (2005), 311-336.
  • G. Mastroianni, M. G. Russo and W. Themistoclakis: The boundedness of the Cauchy singular integral operator in weighted Besov type spaces with uniform norms, Integr. Equ. Oper. Theory, 42, Birkhäuser Verlag Basel (2002), 57-89.
  • G. Mastroianni, W. Themistoclakis: A numerical method for the generalized airfoil equation based on the de la Vallée Poussin interpolation, J. Comput. Appl. Math., 180 (2005), 71-105.
  • C. Micchelli: The saturation class and iterates of Bernstein polynomials, J. Approx. Th., 8 (1973), 1–18
  • G. Monegato: Numerical evaluation of hypersingular integrals, J. Comp. Appl. Math., 50 (1994), 9-31.
  • D. Occorsio, M. G. Russo: A Nyström method for Fredholm integral equations based on equally spaced knots, Filomat, 28 (1) (2014), 49-63.
  • D. Occorsio, M. G. Russo: Bivariate Generalized Bernstein Operators and their application to Fredholm Integral Equations, Publ. Inst. Math. N. S., 100 (114) (2016), 141-162.
  • D. Occorsio, A. C. Simoncelli: How to go from Bézier to Lagrange curves by means of generalized Bézier curves, Facta Univ. Ser. Math. Inform. (Niš), 11 (1996), 101-111.
  • D. Occorsio: Some new properties of Generalized Bernstein polynomials, Stud. Univ. Babe¸s-Bolyai Math, 56 (3) (2011), 147-160.
  • I. Rasa: Iterated Boolean sums of Bernstein and related operators, Rev. Anal. Numér. Théor. Approx, 35 (2006), 111-115.
  • S. Weiwei, W. Jiming: Interpolatory quadrature rules for Hadamard finite-part integrals and their superconvergence, IMA J. Numer. Anal., 28 (2008), 580-597.
  • G. Tachev: From Bernstein polynomials to Lagrange interpolation, Proceedings of 2nd International Conference on Modelling and Development of Intelligent Systems (MDIS 2011), (2011), 192-197.
Toplam 38 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Donatella Occorsıo 0000-0001-9446-4452

Maria Grazia Russo 0000-0002-4078-620X

Woula Themıstoclakıs 0000-0002-6185-1154

Yayımlanma Tarihi 1 Haziran 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 4 Sayı: 2

Kaynak Göster

APA Occorsıo, D., Russo, M. G., & Themıstoclakıs, W. (2021). Some numerical applications of generalized Bernstein operators. Constructive Mathematical Analysis, 4(2), 186-214. https://doi.org/10.33205/cma.868272
AMA Occorsıo D, Russo MG, Themıstoclakıs W. Some numerical applications of generalized Bernstein operators. CMA. Haziran 2021;4(2):186-214. doi:10.33205/cma.868272
Chicago Occorsıo, Donatella, Maria Grazia Russo, ve Woula Themıstoclakıs. “Some Numerical Applications of Generalized Bernstein Operators”. Constructive Mathematical Analysis 4, sy. 2 (Haziran 2021): 186-214. https://doi.org/10.33205/cma.868272.
EndNote Occorsıo D, Russo MG, Themıstoclakıs W (01 Haziran 2021) Some numerical applications of generalized Bernstein operators. Constructive Mathematical Analysis 4 2 186–214.
IEEE D. Occorsıo, M. G. Russo, ve W. Themıstoclakıs, “Some numerical applications of generalized Bernstein operators”, CMA, c. 4, sy. 2, ss. 186–214, 2021, doi: 10.33205/cma.868272.
ISNAD Occorsıo, Donatella vd. “Some Numerical Applications of Generalized Bernstein Operators”. Constructive Mathematical Analysis 4/2 (Haziran 2021), 186-214. https://doi.org/10.33205/cma.868272.
JAMA Occorsıo D, Russo MG, Themıstoclakıs W. Some numerical applications of generalized Bernstein operators. CMA. 2021;4:186–214.
MLA Occorsıo, Donatella vd. “Some Numerical Applications of Generalized Bernstein Operators”. Constructive Mathematical Analysis, c. 4, sy. 2, 2021, ss. 186-14, doi:10.33205/cma.868272.
Vancouver Occorsıo D, Russo MG, Themıstoclakıs W. Some numerical applications of generalized Bernstein operators. CMA. 2021;4(2):186-214.