arXiv subjects

Haiyong Wang

Publications and source records attributed to Haiyong Wang.

At least 19 recordsLinked to original sources

arXiv2026

Superconvergence points of Hermite spectral interpolation

Hermite spectral method plays an important role in the numerical simulation of various partial differential equations (PDEs) on unbounded domains. In this work, we study the superconvergence properties of Hermite spectral interpolation, i.e., interpolation at the zeros of Hermite polynomials in the space spanned by Hermite functions. We identify the points at which the convergence rates of the first- and second-order derivatives of the interpolant converge faster. We further extend the analysis to the Hermite spectral collocation method in solving differential equations and identify the superconvergence points both for function and derivative values. Numerical examples are provided to confirm the analysis of superconvergence points.

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arXiv2026

Hermite spectral approximation for functions with endpoint singularities using exponential transforms

In this paper we introduce Hermite spectral approximation for functions with endpoint singularities using exponential transforms, including single exponential (SE), double exponential (DE) and error function (EF) transforms, and present a comprehensive convergence analysis for these approximations without and with scaling. In the case without scaling, we show that these methods converge at some root-exponential rate. In the case with scaling, we derive optimal scaling factors for each of exponential transforms and show that the convergence rate of Hermite spectral approximation can be significantly improved. Numerical comparisons with sinc method are present and it is shown that Hermite method has comparable or superior accuracy performance when using the same number of terms. Extensions to quadrature and rootfinding algorithm are also discussed.

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arXiv2026

On exponential convergence of Chebyshev polynomial approximation for multivariate analytic functions

This paper presents a new analysis of the Chebyshev projection for multivariate analytic functions, drawing on pluripotential theory. It is proved that in any downward closed convex polynomial space, the Chebyshev projection achieves the same exponential convergence rate as the best polynomial approximation. This result enables a precise quantification of the exponential convergence rate of the Chebyshev projection. The analysis is then extended to several related topics, including tensorized Chebyshev interpolation, tensor product Gauss--Legendre quadrature, Padua interpolation and cubature, and Chebyshev-Galerkin method, with the corresponding exponential convergence rate established in each case. Supporting numerical experiments are provided to validate the theoretical results.

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arXiv2025

Convergence analysis of Hermite approximations for analytic functions

In this paper, we present a rigorous analysis of root-exponential convergence of Hermite approximations, including projection and interpolation methods, for functions that are analytic in an infinite strip containing the real axis and satisfy certain restrictions on the asymptotic behavior at infinity within this strip. Asymptotically sharp error bounds in the weighted and maximum norms are derived. The key ingredients of our analysis are some remarkable contour integral representations for the Hermite coefficients and the remainder of Hermite spectral interpolations. Further extensions to Gauss--Hermite quadrature, Hermite spectral differentiations, generalized Hermite spectral approximations and the scaling factor of Hermite approximation are also discussed. Numerical experiments confirm our theoretical results.

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arXiv2025

Complex generalized Gauss-Radau quadrature rules for Hankel transforms of integer order

Complex Gaussian quadrature rules for oscillatory integral transforms have the advantage that they can achieve optimal asymptotic order. However, their existence for Hankel transform can only be guaranteed when the order of the transform belongs to $[0,1/2]$. In this paper we consider the construction of generalized Gauss-Radau quadrature rules for Hankel transform. We show that, if adding certain value and derivative information at the left endpoint, then complex generalized Gauss-Radau quadrature rules for Hankel transform of integer order can be constructed with theoretical guarantees. Orthogonal polynomials that are closely related to such quadrature rules are investigated and their existence for even degrees is proved. Numerical experiments are presented to confirm our findings.

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arXiv2025

On the deterioration of convergence rate of spectral differentiations for functions with singularities

Spectral differentiations are basic ingredients of spectral methods. In this work, we analyze the pointwise rate of convergence of spectral differentiations for functions containing singularities and show that the deteriorations of the convergence rate at the endpoints, singularities and other points in the smooth region exhibit different patterns. As the order of differentiation increases by one, we show for functions with an algebraic singularity that the convergence rate of spectral differentiation by Jacobi projection deteriorates two orders at both endpoints and only one order at each point in the smooth region. The situation at the singularity is more complicated and the convergence rate either deteriorates two orders or does not deteriorate, depending on the parity of the order of differentiation, when the singularity locates in the interior of the interval and deteriorates two orders when the singularity locates at the endpoint. Extensions to some related problems, such as the spectral differentiation using Chebyshev interpolation, are also discussed. Our findings justify the error localization property of Jacobi approximation and differentiation and provide some new insight into the convergence behavior of Jacobi spectral methods.

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arXiv2023

Convergence analysis of Laguerre approximations for analytic functions

Laguerre spectral approximations play an important role in the development of efficient algorithms for problems in unbounded domains. In this paper, we present a comprehensive convergence rate analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove that Laguerre projection and interpolation methods of degree $n$ converge at the root-exponential rate $O(\exp(-2ρ\sqrt{n}))$ with $ρ>0$ when the underlying function is analytic inside and on a parabola with focus at the origin and vertex at $z=-ρ^2$. As far as we know, this is the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions. Several important applications of our analysis are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules, the scaling factor and the Weeks method for the inversion of Laplace transform, and some sharp convergence rate estimates are derived. Numerical experiments are presented to verify the theoretical results.

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arXiv2023

New error bounds for Legendre approximations of differentiable functions

In this paper we present a new perspective on error analysis of Legendre approximations for differentiable functions. We start by introducing a sequence of Legendre-Gauss-Lobatto polynomials and prove their theoretical properties, such as an explicit and optimal upper bound. We then apply these properties to derive a new and explicit bound for the Legendre coefficients of differentiable functions and establish some explicit and optimal error bounds for Legendre projections in the $L^2$ and $L^{\infty}$ norms. Illustrative examples are provided to demonstrate the sharpness of our new results.

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arXiv2023

Gaussian quadrature rules for composite highly oscillatory integrals

Highly oscillatory integrals of composite type arise in electronic engineering and their calculations is a challenging problem. In this paper, we propose two Gaussian quadrature rules for computing such integrals. The first one is constructed based on the classical theory of orthogonal polynomials and its nodes and weights can be computed efficiently by using tools of numerical linear algebra. We show that the rate of convergence of this rule depends solely on the regularity of the non-oscillatory part of the integrand. The second one is constructed with respect to a sign-changing function and the classical theory of Gaussian quadrature can not be used anymore. We explore theoretical properties of this Gaussian quadrature, including the trajectories of the quadrature nodes and the convergence rate of these nodes to the endpoints of the integration interval, and prove its asymptotic error estimate under suitable hypotheses. Numerical experiments are presented to demonstrate the performance of the proposed methods.

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arXiv2023

Analysis of error localization of Chebyshev spectral approximations

Chebyshev spectral methods are widely used in numerical computations. When the underlying function has a singularity, it has been observed by L. N. Trefethen in 2011 that its Chebyshev interpolants exhibit an error localization property, that is, their errors in a neighborhood of the singularity are obviously larger than elsewhere. In this paper, we first present a pointwise error analysis for Chebyshev projections of functions with a singularity and prove that the rate of convergence of Chebyshev projections of degree $n$ at each point away from the singularity is one power of $n$ faster than that of at the singularity. This gives a rigorous justification for the error localization of Chebyshev projections. We then extend the framework of our analysis to Chebyshev interpolants, Chebyshev spectral differentiations and Legendre projections and justify their error localization using similar arguments. As a result, we find that Chebyshev spectral differentiations converge faster than their best counterparts except in a neighborhood of the singularity and, in the particular case where the singularity is located in the interior of interval, they converge even faster than their best counterparts in the maximum norm.

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arXiv2022

Optimal rates of convergence and error localization of Gegenbauer projections

Motivated by comparing the convergence behavior of Gegenbauer projections and best approximations, we study the optimal rate of convergence for Gegenbauer projections in the maximum norm. We show that the rate of convergence of Gegenbauer projections is the same as that of best approximations under conditions of the underlying function is either analytic on and within an ellipse and $λ\leq0$ or differentiable and $λ\leq1$, where $λ$ is the parameter in Gegenbauer projections. If the underlying function is analytic and $λ>0$ or differentiable and $λ>1$, then the rate of convergence of Gegenbauer projections is slower than that of best approximations by factors of $n^λ$ and $n^{λ-1}$, respectively. An exceptional case is functions with endpoint singularities, for which Gegenbauer projections and best approximations converge at the same rate for all $λ>-1/2$. For functions with interior or endpoint singularities, we provide a theoretical explanation for the error localization phenomenon of Gegenbauer projections and for why the accuracy of Gegenbauer projections is better than that of best approximations except in small neighborhoods of the critical points. Our analysis provides fundamentally new insight into the power of Gegenbauer approximations and related spectral methods.

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arXiv2022

An efficient spectral method for the fractional Schrödinger equation on the real line

The fractional Schrödinger equation (FSE) on the real line arises in a broad range of physical settings and their numerical simulation is challenging due to the nonlocal nature and the power law decay of the solution at infinity. In this paper, we propose a new spectral discretization scheme for the FSE in space based upon Malmquist-Takenaka functions. We show that this new discretization scheme achieves much better performance than existing discretization schemes in the case where the underlying FSE involves the square root of the Laplacian, while in other cases it also exhibits comparable or even better performance. Numerical experiments are provided to illustrate the effectiveness of the proposed method.

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arXiv2020

How much faster does the best polynomial approximation converge than Legendre projection?

We compare the convergence behavior of best polynomial approximations and Legendre and Chebyshev projections and derive optimal rates of convergence of Legendre projections for analytic and differentiable functions in the maximum norm. For analytic functions, we show that the best polynomial approximation of degree $n$ is better than the Legendre projection of the same degree by a factor of $n^{1/2}$. For differentiable functions such as piecewise analytic functions and functions of fractional smoothness, however, we show that the best approximation is better than the Legendre projection by only some constant factors. Our results provide some new insights into the approximation power of Legendre projections.

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arXiv2020

Analysis of multivariate Gegenbauer approximation in the hypercube

In this paper, we are concerned with multivariate Gegenbauer approximation of functions defined in the $d$-dimensional hypercube. Two new and sharper bounds for the coefficients of multivariate Gegenbauer expansion of analytic functions are presented based on two different extensions of the Bernstein ellipse. We then establish an explicit error bound for the multivariate Gegenbauer approximation associated with an $\ell^q$ ball index set in the uniform norm. We also consider the multivariate approximation of functions with finite regularity and derive the associated error bound on the full grid in the uniform norm. As an application, we extend our arguments to obtain some new tight bounds for the coefficients of tensorized Legendre expansions in the context of polynomial approximation of parameterized PDEs.

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arXiv2018

A unified framework for asymptotic analysis and computation of finite Hankel transform

In this paper we present a unified framework for asymptotic analysis and computation of the finite Hankel transform. This framework enables us to derive asymptotic expansions of the transform, including the cases where the oscillator has zeros and stationary points. As a consequence, two efficient and affordable methods for computing the transform numerically are developed and a detailed analysis of their asymptotic error estimate is carried out. Numerical examples are provided to confirm our analysis.

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arXiv2018

A new and sharper bound for Legendre expansion of differentiable functions

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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arXiv2017

Jacobi polynomials on the Bernstein ellipse

In this paper, we are concerned with Jacobi polynomials $P_n^{(α,β)}(x)$ on the Bernstein ellipse with motivation mainly coming from recent studies of convergence rate of spectral interpolation. An explicit representation of $P_n^{(α,β)}(x)$ is derived in the variable of parametrization. This formula further allows us to show that the maximum value of $\left|P_n^{(α,β)}(z)\right|$ over the Bernstein ellipse is attained at one of the endpoints of the major axis if $α+β\geq -1$. For the minimum value, we are able to show that for a large class of Gegenbauer polynomials (i.e., $α=β$), it is attained at two endpoints of the minor axis. These results particularly extend those previously known only for some special cases. Moreover, we obtain a more refined asymptotic estimate for Jacobi polynomials on the Bernstein ellipse.

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arXiv2016

On the optimal estimates and comparison of Gegenbauer expansion coefficients

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.307-327]. 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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