Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey

Mastroianni, G.; Occorsio, Donatella

doi:10.1016/s0377-0427(00)00557-4

Cited by 108 publications

(44 citation statements)

References 28 publications

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“…We have the following result on the Lebesgue constant for the Lagrange interpolation polynomials associated with the zeros of the Jacobi polynomials; see, e.g., [18]. …”

Section: Then For the Jacobi Gauss And Jacobi Gauss-radau Integrationmentioning

confidence: 99%

Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel

2010

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Abstract. In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of the second kind with a weakly singular kernel. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [−1, 1], so that the solution of the new equation possesses better regularity and the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high-order accuracy for the approximation, the integral term in the resulting equation is approximated by using Jacobi spectral quadrature rules. The convergence analysis of this novel method is based on the Lebesgue constants corresponding to the Lagrange interpolation polynomials, polynomial approximation theory for orthogonal polynomials and operator theory. The spectral rate of convergence for the proposed method is established in the L ∞ -norm and the weighted L 2 -norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

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“…We have the following result on the Lebesgue constant for the Lagrange interpolation polynomials associated with the zeros of the Jacobi polynomials; see, e.g., [18]. …”

Section: Then For the Jacobi Gauss And Jacobi Gauss-radau Integrationmentioning

confidence: 99%

Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel

2010

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“…Lemma 3.5 (see [9]). For every bounded function u, there exists a constant C, independent of u such that…”

Section: Some Useful Lemmasmentioning

confidence: 99%

“…Lemma 3.6 (see [9]). Assume that {F j (x)} N j=0 are the N -th degree Lagrange basis polynomials associated with the Gauss points of the Jacobi polynomials.…”

Section: Some Useful Lemmasmentioning

confidence: 99%

Jacobi Spectral Galerkin Methods for Volterra Integral Equations With Weakly Singular Kernel

Yang¹

2016

Bulletin of the Korean Mathematical Society

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Abstract. We propose and analyze spectral and pseudo-spectral JacobiGalerkin approaches for weakly singular Volterra integral equations (VIEs). We provide a rigorous error analysis for spectral and pseudospectral Jacobi-Galerkin methods, which show that the errors of the approximate solution decay exponentially in L ∞ norm and weighted L 2 -norm. The numerical examples are given to illustrate the theoretical results.

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“…From [22], we have the following result on the Lebesgue constant for Lagrange interpolation based on the zeros of the Jacobi polynomials. …”

Section: Lemma 31 (Estimates For Interpolation Error) Assume That ∂ mentioning

confidence: 99%

Jacobi-spectral method for integro-delay differential equations with weakly singular kernels

Ali

2015

Turk J Math

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We present a numerical solution to the integro-delay differential equation with weakly singular kernels with the delay function θ(t) vanishing at the initial point of the given interval [0, T ] ( θ(t) = qt, 0 < q < 1) . In order to fully use the Jacobi orthogonal polynomial theory, we use some function and variable transformation to change the intergrodelay differential equation into a new equation defined on the standard interval [−1, 1] . A Gauss-Jacobi quadrature formula is used to evaluate the integral term. The spectral rate of convergence is provided in infinity norm under the assumption that the solution of the given equation is sufficiently smooth. For validation of the theoretical exponential rate of convergence of our method, we provide some numerical examples.

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Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey

Cited by 108 publications

References 28 publications

Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel

Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel

Jacobi Spectral Galerkin Methods for Volterra Integral Equations With Weakly Singular Kernel

Jacobi-spectral method for integro-delay differential equations with weakly singular kernels

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