Filtered integration rules for finite Hilbert transforms

A product quadrature rule, based on the filtered de la Vallée Poussin polynomial approximation, is proposed for evaluating the finite Hilbert transform in [−1, 1]. Convergence results are stated in weighted uniform norm for functions belonging to suitable Besov type subspaces. Several numerical tests are provided, also comparing the rule with other formulas known in literature.

[1]  D. W. Arthur,et al.  Methods of numerical Integration (2nd edition), by Philip J. Davis and Philip Rabinowitz. Pp 612. £36·50. 1984. ISBN 0-12-206360-0 (Academic Press) , 1986, The Mathematical Gazette.

[2]  Giovanni Monegato,et al.  On the weights of certain quadratures for the numerical evaluation of Cauchy principal value integrals and their derivatives , 1986 .

[3]  Giuseppe Mastroianni,et al.  LAGRANGE INTERPOLATION IN SOME WEIGHTED UNIFORM SPACES , 2012 .

[4]  P. Keast,et al.  A sinc quadrature subroutine for Cauchy principal value integrals , 1999 .

[5]  Woula Themistoclakis,et al.  Weighted L1 approximation on [-1, 1] via discrete de la Vallée Poussin means , 2017, Math. Comput. Simul..

[6]  G. Milovanović,et al.  Interpolation Processes: Basic Theory and Applications , 2008 .

[7]  M. R. Capobianco,et al.  Uniform convergence of the collocation method for Prandtl's Integro-differential equation , 2000, The ANZIAM Journal.

[8]  B. Silbermann,et al.  Numerical Analysis for Integral and Related Operator Equations , 1991 .

[9]  G. Monegato.  Quadrature formulas for functions with poles near the interval of integration , 1986 .

[10]  Woula Themistoclakis,et al.  Uniform weighted approximation on the square by polynomial interpolation at Chebyshev nodes , 2020, Appl. Math. Comput..

[11]  Giuseppe Mastroianni,et al.  On the convergence of an interpolatory product rule for evaluating Cauchy principal value integrals , 1987 .

[12]  Giovanni Monegato,et al.  The numerical evaluation of one-dimensional Cauchy principal value integrals , 1982, Computing.

[13]  Giovanni Monegato,et al.  Quadrature rules for Prandtl's integral equation , 1986, Computing.

[14]  Woula Themistoclakis,et al.  A numerical method for the generalized airfoil equation based on the de la Vallée Poussin interpolation , 2005 .

[15]  D. Occorsio,et al.  Some remarks on filtered polynomial interpolation at Chebyshev nodes , 2021, ArXiv.

[16]  V. Totik,et al.  Moduli of smoothness , 1987 .

[17]  I. M. Pyshik,et al.  Table of integrals, series, and products , 1965 .

[18]  Woula Themistoclakis,et al.  On the filtered polynomial interpolation at Chebyshev nodes , 2020, ArXiv.

[19]  Woula Themistoclakis,et al.  Uniform approximation on [−1, 1] via discrete de la Vallée Poussin means , 2012, Numerical Algorithms.

[20]  Ian H. Sloan,et al.  Numerical Solution of the Generalized Airfoil Equation for an Airfoil with a Flap , 1997 .

[21]  Tatsuo Torii,et al.  An automatic quadrature for Cauchy principal value integrals , 1991 .

[22]  A. I. Kalandiya.  Mathematical Methods of Two-Dimensional Elasticity. , 1975 .

[23]  Kerstin Vogler,et al.  Table Of Integrals Series And Products , 2016 .

[24]  R. Kanwal.  Linear Integral Equations , 1925, Nature.

[25]  U. Luther.  Cauchy Singular Integral Operators in Weighted Spaces of Continuous Functions , 2005 .

[26]  Kai Diethelm,et al.  Uniform convergence of optimal order quadrature rules for Cauchy principal value integrals , 1994 .

[27]  U. Luther,et al.  Boundedness of the Hilbert transformation in some weighted Besov type spaces , 2000 .

[28]  G. Mastroianni,et al.  Numerical Methods for Cauchy Singular Integral Equations in Spaces of Weighted Continuous Functions , 2005 .

[29]  Maria Carmela De Bonis,et al.  Quadrature methods for integro-differential equations of Prandtl's type in weighted spaces of continuous functions , 2020, Appl. Math. Comput..

[30]  Woula Themistoclakis,et al.  Generalized de la Vallée Poussin approximations on [−1, 1] , 2017, Numerical Algorithms.

[31]  B. Silbermann,et al.  A fast algorithm for solving the generalized airfoil equation , 1992 .

[32]  W. Themistoclakis,et al.  Filtered interpolation for solving Prandtl’s integro-differential equations , 2020, Numerical Algorithms.

[33]  Peter Junghanns,et al.  Cauchy singular integral equations in spaces of continuous functions and methods for their numerical solution , 1997 .

[34]  Remarks on Besov spaces and best polynomial approximation , 1988 .

[35]  F. Filbir,et al.  Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced Nodes , 2020, Mathematics.