A Padé-based algorithm for overcoming the Gibbs phenomenon

Truncated Fourier series and trigonometric interpolants converge slowly for functions with jumps in value or derivatives. The standard Fourier–Padé approximation, which is known to improve on the convergence of partial summation in the case of periodic, globally analytic functions, is here extended to functions with jumps. The resulting methods (given either expansion coefficients or function values) exhibit exponential convergence globally for piecewise analytic functions when the jump location(s) are known. Implementation requires just the solution of a linear system, as in standard Padé approximation. The new methods compare favorably in experiments with existing techniques.     

The resolution of the Gibbs phenomenon for spherical harmonics

Spherical harmonics have been important tools for solving geophysical and astrophysical problems. Methods have been developed to effectively implement spherical harmonic expansion approximations. However, the Gibbs phenomenon was already observed by Weyl for spherical harmonic expansion approximations to functions with discontinuities, causing undesirable oscillations over the entire sphere.

Recently, methods for removing the Gibbs phenomenon for one-dimensional discontinuous functions have been successfully developed by Gottlieb and Shu. They proved that the knowledge of the first expansion coefficients (either Fourier or Gegenbauer) of a piecewise analytic function is enough to recover an exponentially convergent approximation to the point values of in any subinterval in which the function is analytic.

Here we take a similar approach, proving that knowledge of the first spherical harmonic coefficients yield an exponentially convergent approximation to a spherical piecewise smooth function in any subinterval , where the function is analytic. Thus we entirely overcome the Gibbs phenomenon.

     

Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations

Summary. We prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation by a (periodic) boundary value approach. This equation arises, e.g., in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations. We obtain convergence results for the standard collocation algorithm and for two variants. In particular, estimates of the difference between the collocation solution and the true solution are derived. For the standard collocation scheme the convergence results are “unconditional”, that is, they do not require mesh-ratio restrictions. Numerical results that support the theoretical findings are also given. Received June 9, 2000 / Revised version received December 14, 2000 / Published online October 17, 2001     

Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon

We introduce a simple and efficient method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we establish, provided the dimension of the reconstruction space is chosen suitably in relation to the number of samples, this procedure can be implemented in a completely numerically stable manner. Moreover, the accuracy of the resulting approximation is determined solely by the choice of reconstruction basis, meaning that reconstruction vectors can be readily tailored to the particular problem at hand.An important example of this approach is the accurate recovery of a piecewise analytic function from its first few Fourier coefficients. Whilst the standard Fourier projection suffers from the Gibbs phenomenon, by reconstructing in a piecewise polynomial basis we obtain an approximation with root-exponential accuracy in terms of the number of Fourier samples and exponential accuracy in terms of the degree of the reconstruction. Numerical examples illustrate the advantage of this approach over other existing methods.     

Error autocorrection in rational approximation and interval estimates. [A survey of results.]

The error autocorrection effect means that in a calculation all the intermediate errors compensate each other, so the final result is much more accurate than the intermediate results. In this case standard interval estimates (in the framework of interval analysis including the so-called a posteriori interval analysis of Yu. Matijasevich) are too pessimistic. We shall discuss a very strong form of the effect which appears in rational approximations to functions. The error autocorrection effect occurs in all efficient methods of rational approximation (e.g., best approxmations, Padé approximations, multipoint Padé approximations, linear and nonlinear Padé-Chebyshev approximations, etc.), where very significant errors in the approximant coefficients do not affect the accuracy of this approximant. The reason is that the errors in the coefficients of the rational approximant are not distributed in an arbitrary way, but form a collection of coefficients for a new rational approximant to the same approximated function. The understanding of this mechanism allows to decrease the approximation error by varying the approximation procedure depending on the form of the approximant. Results of computer experiments are presented. The effect of error autocorrection indicates that variations of an approximated function under some deformations of rather a general type may have little effect on the corresponding rational approximant viewed as a function (whereas the coefficients of the approximant can have very significant changes). Accordingly, while deforming a function for which good rational approximation is possible, the corresponding approximant’s error can rapidly increase, so the property of having good rational approximation is not stable under small deformations of the approximated functions. This property is “individual”, in the sense that it holds for specific functions.     

Iterative methods based on spline approximations to detect discontinuities from Fourier data

Recently, spline approximations have been proposed for the reconstruction of piecewise smooth functions from Fourier data. That approach makes possible to retrieve the functions from their Fourier coefficients for any given degree of accuracy when the discontinuity points are known. In this paper we present iterative methods based on those spline approximations, for several degrees, to find locations and amplitudes of the jumps of a piecewise smooth function, given its Fourier coefficients. We also present numerical experiments comparing with different previous approaches.     

A multi-interval Chebyshev collocation approach for the stability of periodic delay systems with discontinuities

This paper investigates the stability of periodic delay systems with non-smooth coefficients using a multi-interval Chebyshev collocation approach (MIC). In this approach, each piecewise continuous interval is expanded in a Chebyshev basis of the first order. The boundaries of these intervals are placed at the points of discontinuity to recover the fast convergence properties of spectral methods. Stability is examined for a set of case studies that contain the complexities of periodic coefficients, delays and discontinuities. The new approach is also compared to the conventional Chebyshev collocation method.     

Approximating the jump discontinuities of a function by its Fourier-Jacobi coefficients

In the present paper we generalize Eckhoff's method, i.e., the method for approximating the locations of discontinuities and the associated jumps of a piecewise smooth function by means of its Fourier-Chebyshev coefficients.

A new method enables us to approximate the locations of discontinuities and the associated jumps of a discontinuous function, which belongs to a restricted class of the piecewise smooth functions, by means of its Fourier-Jacobi coefficients for arbitrary indices. Approximations to the locations of discontinuities and the associated jumps are found as solutions of algebraic equations. It is shown as well that the locations of discontinuities and the associated jumps are recovered exactly for piecewise constant functions with a finite number of discontinuities.

In addition, we study the accuracy of the approximations and present some numerical examples.

     

A modified Lyapunov method and its applications to ODE

Here, we propose a method to obtain local analytic approximate solutions of ordinary differential equations with variable coefficients, or even some nonlinear equations, inspired in the Lyapunov method, where instead of polynomial approximations, we use truncated Fourier series with variable coefficients as approximate solutions. In the case of equations admitting periodic solutions, an averaging over the coefficients gives global solutions. We show that, under some restrictive condition, the method is equivalent to the Picard-Lindelöf method. After some numerical experiments showing the efficiency of the method, we apply it to equations of interest in physics, in which we show that our method possesses an excellent precision even with low iterations.     

Nevanlinna theory, diophantine approximation, and numerical analysis

This article treats the problem of the approximation of an analytic function f on the unit disk by rational functions having integral coefficients, with the goodness of each approximation being judged in terms of the maximum of the absolute values of the coefficients of the rational function. This relates to the more usual approximation by a rational function in that it could imply how many decimal places are needed when applying a particularly good rational function approximation having non-integrad coefficients. It is shown how to obtain “good” approximations of this type and it is also shown how under certain circumstances “very good” bounds are not possible. As in diophantine approximation this means that many merely “good” approximations do exist, which may be the preferable case. The existence or nonexistence of “very good” approximations is closely related to the diophantine approximation of the first nonzero power series coefficient of at z=0. Nevanlinna theory methods are used in the proofs.     

Approximation of classes of periodic functions with small smoothness

We prove that the approximations of classes of periodic functions with small smoothness in the metrics of the spaces C and L by different linear summation methods for Fourier series are asymptotically equal to the least upper bounds of the best approximations of these classes by trigonometric polynomials of degree not higher than (n - 1). We establish that the Fejér method is asymptotically the best among all positive linear approximation methods for these classes.     

Effective numerical evaluation of the double Hilbert transform

In this paper, we propose two methods to compute the double Hilbert transform of periodic functions. First, we establish the quadratic formula of trigonometric interpolation type for double Hilbert transform and obtain an estimation of the remainder. We call this method 2D mechanical quadrature method (2D-MQM). Numerical experiments show that 2D-MQM outperforms the library function “hilbert” in Matlab when the values of the functions being handled are very large or approach to infinity. Second, we propose a complex analytic method to calculate the double Hilbert transform, which is based on the 2D adaptive Fourier decomposition, and the method is called as 2D-HAFD. In contrast to the pointwise approximation, 2D-HAFD provides explicit rational functional approximations and is valid for all signals of finite energy.     

Continuous collocation approximations to solutions of first kind Volterra equations

In this paper we give necessary and sufficient conditions for convergence of continuous collocation approximations of solutions of first kind Volterra integral equations. The results close some longstanding gaps in the theory of polynomial spline collocation methods for such equations. The convergence analysis is based on a Runge-Kutta or ODE approach.

     

An iterative substructuring method for Maxwell's equations in two dimensions

Iterative substructuring methods, also known as Schur complement methods, form an important family of domain decomposition algorithms. They are preconditioned conjugate gradient methods where solvers on local subregions and a solver on a coarse mesh are used to construct the preconditioner. For conforming finite element approximations of , it is known that the number of conjugate gradient steps required to reduce the residual norm by a fixed factor is independent of the number of substructures, and that it grows only as the logarithm of the dimension of the local problem associated with an individual substructure. In this paper, the same result is established for similar iterative methods for low-order Nédélec finite elements, which approximate in two dimensions. Results of numerical experiments are also provided.

     

Thermodynamic formalism for countable to one Markov systems

For countable to one transitive Markov systems we establish thermodynamic formalism for non-Hölder potentials in nonhyperbolic situations. We present a new method for the construction of conformal measures that satisfy the weak Gibbs property for potentials of weak bounded variation and show the existence of equilibrium states equivalent to the weak Gibbs measures. We see that certain periodic orbits cause a phase transition, non-Gibbsianness and force the decay of correlations to be slow. We apply our results to higher-dimensional maps with indifferent periodic points.

     

有理曲线和曲面的分片多项式逼近

本文利用摄动的思想,以摄动有理曲线（曲面）的系数的无穷模作为优化目标,给出了用多项式曲线（曲面）逼近有理曲线（曲面）的一种新方法．同以前的各种方法相比,该方法不仅收敛而且具有更快的收敛速度,并且可以与细分技术相结合,得到有理曲线与曲面的整体光滑、分片多项式的逼近．     

Precise Asymptotic Approximations for Kernels Corresponding to Lévy Processes

By using basic complex analysis techniques, we obtain precise asymptotic approximations for kernels corresponding to symmetric α-stable processes and their fractional derivatives. We use the deep connection between the decay of kernels and singularities of the Mellin transforms. The key point of the method is to transform the multi-dimensional integral to the contour integral representation. We then express the integrand as a combination of gamma functions so that we can easily find all poles of the integrand. We obtain various asymtotics of the kernels by using Cauchys Residue Theorem with shifting contour integration. As a byproduct, exact coefficients are also obtained. We apply this method to general Lévy processes whose characteristic functions are radial and satisfy some regularity and size conditions. Our approach is based on the Fourier analytic point of view.     

Nonlinear inversion of a band-limited Fourier transform

We consider the problem of reconstructing a compactly supported function with singularities either from values of its Fourier transform available only in a bounded interval or from a limited number of its Fourier coefficients. Our results are based on several observations and algorithms in [G. Beylkin, L. Monzón, On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal. 19 (1) (2005) 17–48]. We avoid both the Gibbs phenomenon and the use of windows or filtering by constructing approximations to the available Fourier data via a short sum of decaying exponentials. Using these exponentials, we extrapolate the Fourier data to the whole real line and, on taking the inverse Fourier transform, obtain an efficient rational representation in the spatial domain. An important feature of this rational representation is that the positions of its poles indicate location of singularities of the function. We consider these representations in the absence of noise and discuss the impact of adding white noise to the Fourier data. We also compare our results with those obtained by other techniques. As an example of application, we consider our approach in the context of the kernel polynomial method for estimating density of states (eigenvalues) of Hermitian operators. We briefly consider the related problem of approximation by rational functions and provide numerical examples using our approach.     

Convergence of Collocation Method with Delta Functions for Integral Equations of First Kind

Integral equations of first kind with periodic kernels arising in solving partial differential equations by interior source methods are considered. Existence and uniqueness of solution in appropriate spaces of linear analytic functionals is proved. Rate of convergence of collocation method with Dirac’s delta-functions as the trial functions is obtained in case of uniform meshes. In case of an analytic kernel the convergence rate is exponential.     

Lower bound for a linear form

By using Padé approximations of the first kind, we obtain a lower bound for the absolute value of a linear form with integer coefficients from the values of polylogarithmic functions at rational points. This estimate takes into account the growth of all coefficients of the linear form. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 617–623, October, 1999.