Sharp Error Bounds for Jacobi  Expansions and Gegenbauer--Gauss Quadrature of Analytic Functions

Zhao, Xiaodan; Wang, Li-Lian; Xie, Zhongliang

doi:10.1137/12089421x

Cited by 47 publications

(34 citation statements)

References 37 publications

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“…that is provided in [41,Theorem 3.2] for the Gauss-Gegenbauer quadrature error for a function v ∈ A ρ . Letting v = fC (s+1/2) j with j ≤ n, equation (5.9) and (4.20) yield (5.10) |f…”

Section: High Order Numerical Methodsmentioning

confidence: 99%

“…In order to obtain suitable coefficient bounds, we note that, since f ∈ A ρ , there indeed exists ρ 2 > ρ such that f ∈ A ρ2 . It follows [41] that the Gegenbauer coefficients decay exponentially. More precisely, for a certain constant C we have the estimate In order to the adequately account for the growth of the Gegenbauer polynomials, on the other hand, we consider the estimate Let now ρ 1 ∈ [ρ, ρ 2 ).…”

Section: Regularity Theorymentioning

confidence: 99%

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Regularity theory and high order numerical methods for the (1D)-fractional Laplacian

et al. 2017

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Abstract. This paper presents regularity results and associated high-order numerical methods for one-dimensional Fractional-Laplacian boundary-value problems. On the basis of a factorization of solutions as a product of a certain edge-singular weight ω times a "regular" unknown, a characterization of the regularity of solutions is obtained in terms of the smoothness of the corresponding right-hand sides. In particular, for right-hand sides which are analytic in a Bernstein Ellipse, analyticity in the same Bernstein Ellipse is obtained for the "regular" unknown. Moreover, a sharp Sobolev regularity result is presented which completely characterizes the co-domain of the Fractional-Laplacian operator in terms of certain weighted Sobolev spaces introduced in (Babuška and Guo, SIAM J. Numer. Anal. 2002). The present theoretical treatment relies on a full eigendecomposition for a certain weighted integral operator in terms of the Gegenbauer polynomial basis. The proposed Gegenbauer-based Nyström numerical method for the Fractional-Laplacian Dirichlet problem, further, is significantly more accurate and efficient than other algorithms considered previously. The sharp error estimates presented in this paper indicate that the proposed algorithm is spectrally accurate, with convergence rates that only depend on the smoothness of the right-hand side. In particular, convergence is exponentially fast (resp. faster than any power of the mesh-size) for analytic (resp. infinitely smooth) right-hand sides. The properties of the algorithm are illustrated with a variety of numerical results.

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“…that is provided in [41,Theorem 3.2] for the Gauss-Gegenbauer quadrature error for a function v ∈ A ρ . Letting v = fC (s+1/2) j with j ≤ n, equation (5.9) and (4.20) yield (5.10) |f…”

Section: High Order Numerical Methodsmentioning

confidence: 99%

Section: Regularity Theorymentioning

confidence: 99%

Regularity theory and high order numerical methods for the (1D)-fractional Laplacian

et al. 2017

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“…Indeed, it is useful not only in understanding the rate of convergence of Legendre expansion but useful also in estimating the degree of the Legendre polynomial approximation to f (x) within a given accuracy. When f (x) is analytic in a neighborhood of the interval [−1, 1], we note that the estimate of the Legendre coefficients, or more generally, the Gegenbauer and Jacobi coefficients, has been studied in [5,6,7,8]. The analysis in those references is either built on the connection relation between Chebyshev and Legendre polynomials [5,7,8] or built on the contour integral expression of the Legendre coefficients [6].…”

Section: Introductionmentioning

confidence: 99%

A new and sharper bound for Legendre expansion of differentiable functions

Wang

2018

Applied Mathematics Letters

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In this paper, we provide a new and sharper bound for the Legendre coefficients of differentiable functions and then derive a new error bound of the truncated Legendre series in the uniform norm. The key idea of proof relies on integration by parts and a sharp Bernstein-type inequality for the Legendre polynomial. An illustrative example is provided to demonstrate the sharpness of our new results.

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“…(4.3)]). More recently, this issue was considered in [32,33] and some sharper estimates were given. In these works, the main idea is to express the Gegenbauer coefficient a λ n as an infinite series either by using the Chebyshev expansion of the first kind of f or the Cauchy integral formula together with the generating function of the Chebyshev polynomial of the second kind, and then estimate the derived infinite series term by term.…”

Section: Introductionmentioning

confidence: 99%

On the Optimal Estimates and Comparison of Gegenbauer Expansion Coefficients

Wang¹

2016

SIAM J. Numer. Anal.

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In this paper, we study optimal estimates and comparison of the coefficients in the Gegenbauer series expansion. We propose an alternative derivation of the contour integral representation of the Gegenbauer expansion coefficients which was recently derived by Cantero and Iserles [SIAM J. Numer. Anal., 50 (2012), pp. . With this representation, we show that optimal estimates for the Gegenbauer expansion coefficients can be derived, which in particular includes Legendre coefficients as a special case. Further, we apply these estimates to establish some rigorous and computable bounds for the truncated Gegenbauer series. In addition, we compare the decay rates of the Chebyshev and Legendre coefficients. For functions whose singularity is outside or at the endpoints of the expansion interval, asymptotic behaviour of the ratio of the nth Legendre coefficient to the nth Chebyshev coefficient is given, which provides us an illuminating insight for the comparison of the spectral methods based on Legendre and Chebyshev expansions.

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Sharp Error Bounds for Jacobi Expansions and Gegenbauer--Gauss Quadrature of Analytic Functions

Cited by 47 publications

References 37 publications

Regularity theory and high order numerical methods for the (1D)-fractional Laplacian

Regularity theory and high order numerical methods for the (1D)-fractional Laplacian

A new and sharper bound for Legendre expansion of differentiable functions

On the Optimal Estimates and Comparison of Gegenbauer Expansion Coefficients

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