A method for the practical evaluation of the Hilbert transform on the real line

DOI10.1016/S0377-0427(99)00212-5zbMath0944.65139WikidataQ126297358 ScholiaQ126297358MaRDI QIDQ1964075

Kai Diethelm

Publication date: 24 September 2000

Published in: Journal of Computational and Applied Mathematics (Search for Journal in Brave)

zbMATH Keywords

performancenumerical exampleserror analysisHilbert transformquadrature formulanumerical stabilityHermite weight functionpolynomial interpolationCauchy principal value integral

Mathematics Subject Classification ID

Special integral transforms (Legendre, Hilbert, etc.) (44A15) Numerical methods for integral transforms (65R10) Approximate quadratures (41A55) Numerical quadrature and cubature formulas (65D32) Complexity and performance of numerical algorithms (65Y20)

Related Items (8)

Constructing solutions of Cauchy type integral equations by using four kinds of basis ⋮ On uniform approximations to hypersingular finite-part integrals ⋮ Uniform approximations to finite Hilbert transform and its derivative. ⋮ Numerical steepest descent method for Hankel type of hypersingular oscillatory integrals in electromagnetic scattering problems ⋮ BOUNDEDNESS AND UNIFORM NUMERICAL APPROXIMATION OF THE WEIGHTED HILBERT TRANSFORM ON THE REAL LINE ⋮ Clenshaw-Curtis-type quadrature rule for hypersingular integrals with highly oscillatory kernels ⋮ Convergence and stability of a new quadrature rule for evaluating Hilbert transform ⋮ Numerical evaluation of Hilbert transforms for oscillatory functions: A convergence accelerator approach

Uses Software

• Mathematica
• QUADPACK
• nag
• Algorithm 680

Cites Work

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• Computing the Hilbert transform of a Jacobi weight function
• The numerical evaluation of one-dimensional Cauchy principal value integrals
• Approximations via Whittaker's cardinal function
• Bounds for Lebesgue functions for Freud weights
• New error bounds for modified quadrature formulas for Cauchy principal value integrals
• Functions of the second kind for Freud weights and series expansions of Hilbert transforms
• Interlacing properties of the zeros of the orthogonal polynomials and approximation of the Hilbert transform
• Gaussian quadrature formulae of the third kind for Cauchy principal value integrals: Basic properties and error estimates
• Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals
• Piecewise-Polynomial Quadratures for Cauchy Singular Integrals
• Algorithm 680: evaluation of the complex error function
• Computing the Hilbert Transform on the Real Line
• An Automatic Quadrature for Cauchy Principal Value Integrals
• Anwendung der Rechteckregel auf die reelle Hilberttransformation mit unendlichem Intervall
• Jackson and smoothness theorems for Freud weights in \(L_ p (0<p\leq\infty)\)

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