Fourier and barycentric formulae for equidistant Hermite trigonometric interpolation

We consider the Hermite trigonometric interpolation problem of order 1 for equidistant nodes, i.e., the problem of finding a trigonometric polynomial t that interpolates the values of a function and of its derivative at equidistant points. We give a formula for the Fourier coefficients of t in terms of those of the two classical trigonometric polynomials interpolating the values and those of the derivative separately. This formula yields the coefficients with a single FFT. It also gives an aliasing formula for the error in the coefficients which, on its turn, yields error bounds and convergence results for differentiable as well as analytic functions. We then consider the Lagrangian formula and eliminate the unstable factor by switching to the barycentric formula. We also give simplified formulae for even and odd functions, as well as consequent formulae for Hermite interpolation between Chebyshev points.     

On Trigonometric Blending Interpolation and Cubature Formulae

We discuss error representations for Hermite-Lagrange trigonometric interpolation introduced in Dryanov and Petrov (Interpolation and L ₁-approximation by trigonometric polynomials and blending functions, J. Approx. Theory 164, 1049–1064 (2012)) and obtain one-sided trigonometric quadratures for approximate integration of one-dimensional integrals. Next, we study error representations of multivariate Hermite-Lagrange transfinite trigonometric interpolation and derive one-sided trigonometric blending interpolants to multivariate functions, under some restrictions. Then, we construct one-sided transfinite cubature formulae for approximate integration of multivariate integrals. We construct also cubature formulae with positive coefficients, based on line integrals and exact in a vector space of trigonometric blending functions with prescribed order.     

Weighted interpolation for equidistant nodes

Weighted Lagrange interpolation is proposed for solving Lagrange interpolation problems on equidistant or almost equidistant data. Good condition numbers are found in the case of rational interpolants whose denominator has degree about twice the number of data to be interpolated. Since the degree of the denominator is higher than that of the numerator, simple functions like constants and linear polynomials will not be reproduced. Furthermore, the interpolant cannot be expressed by a barycentric formula. As a counterpart, the interpolation algorithm is simple and leads to small Lebesgue constants.     

Applications of?the Cosecant and Related Numbers

Power series expansions for cosecant and related functions together with a vast number of applications stemming from their coefficients are derived here. The coefficients for the cosecant expansion can be evaluated by using: (1) numerous recurrence relations, (2) expressions resulting from the application of the partition method for obtaining a power series expansion and (3) the result given in Theorem 3. Unlike the related Bernoulli numbers, these rational coefficients, which are called the cosecant numbers and are denoted by c _k , converge rapidly to zero as k????. It is then shown how recent advances in obtaining meaningful values from divergent series can be modified to determine exact numerical results from the asymptotic series derived from the Laplace transform of the power series expansion for tcsc?(at). Next the power series expansion for secant is derived in terms of related coefficients known as the secant numbers d _k . These numbers are related to the Euler numbers and can also be evaluated by numerous recurrence relations, some of which involve the cosecant numbers. The approaches used to obtain the power series expansions for these fundamental trigonometric functions in addition to the methods used to evaluate their coefficients are employed in the derivation of power series expansions for integer powers and arbitrary powers of the trigonometric functions. Recurrence relations are of limited benefit when evaluating the coefficients in the case of arbitrary powers. Consequently, power series expansions for the Legendre-Jacobi elliptic integrals can only be obtained by the partition method for a power series expansion. Since the Bernoulli and Euler numbers give rise to polynomials from exponential generating functions, it is shown that the cosecant and secant numbers gives rise to their own polynomials from trigonometric generating functions. As expected, the new polynomials are related to the Bernoulli and Euler polynomials, but they are found to possess far more interesting properties, primarily due to the convergence of the coefficients. One interesting application of the new polynomials is the re-interpretation of the Euler-Maclaurin summation formula, which yields a new regularisation formula.     

On the Lebesgue constant of barycentric rational interpolation at equidistant nodes

Recent results reveal that the family of barycentric rational interpolants introduced by Floater and Hormann is very well-suited for the approximation of functions as well as their derivatives, integrals and primitives. Especially in the case of equidistant interpolation nodes, these infinitely smooth interpolants offer a much better choice than their polynomial analogue. A natural and important question concerns the condition of this rational approximation method. In this paper we extend a recent study of the Lebesgue function and constant associated with Berrut’s rational interpolant at equidistant nodes to the family of Floater–Hormann interpolants, which includes the former as a special case.     

Bernstein Polynomial Collocation Method for Elliptic Boundary Value Problems

We present a summary of recent developments in application of Bernstein polynomials to solution of elliptic boundary value problems with a pseudospectral method. Solution is approximated using Benstein polynomial interpolant defined at points of the extrema of Chebyshev polynomials i.e. the Chebyshev-Gauss-Lobatto (CGL) nodes. This approach brings impovement comparing to the Bernstein interpolation at equidistant nodes we used previously [1]. We show that this approach leads to spectral convergence and accuracy comparable to that of pseudospectral methods with orthogonal polynomials (Chebyshev, Legendre). The algorithm is implemented in open source library bernstein-poly , which is available online. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lagrange基函数的复矩阵有理插值及连分式插值

1引言 矩阵有理插值问题与系统线性理论中的模型简化问题和部分实现问题有着紧密的联系~[1][2]，在矩阵外推方法中也常常涉及线性或有理矩阵插值问题~[3]。按照文~[1]的阐述。目前已经研究的矩阵有理插值问题包括矩阵幂级数和Newton-Pade逼近。Hade逼近，联立Pade逼近，M-Pade逼近，多点Pade逼近等。显然，上述各种形式的矩阵Pade逼上梁山近是矩     

On general Hermite trigonometric interpolation

Summary A sequence of general Hermite trigonometric interpolation polynomials with equidistant interpolation points is given. Integrating these interpolation formulae a sequence of quadrature formulae for the integration of periodic functions is obtained. Derivative-free remainders are stated for these interpolation and quadrature formulae.This work was done at the Max-Planck-Institut für Physik und Astrophysik, München.     

Convergence rates of derivatives of a family of barycentric rational interpolants

In polynomial and spline interpolation the k-th derivative of the interpolant, as a function of the mesh size h, typically converges at the rate of O(hd+1−k) as h→0, where d is the degree of the polynomial or spline. In this paper we establish, in the important cases k=1,2, the same convergence rate for a recently proposed family of barycentric rational interpolants based on blending polynomial interpolants of degree d.     

Admissible slopes for monotone and convex interpolation

Summary In many applications, interpolation of experimental data exhibiting some geometric property such as nonnegativity, monotonicity or convexity is unacceptable unless the interpolant reflects these characteristics. This paper identifies admissible slopes at data points of variousC ¹ interpolants which ensure a desirable shape. We discuss this question, in turn for the following function classes commonly used for shape preserving interpolations: monotone polynomials,C ¹ monotone piecewise polynomials, convex polynomials, parametric cubic curves and rational functions.     

Discrete Fourier Analysis on Fundamental Domain and Simplex of A d Lattice in d-Variables

A discrete Fourier analysis on the fundamental domain Ω _d of the d-dimensional lattice of type A _d is studied, where Ω₂ is the regular hexagon and Ω₃ is the rhombic dodecahedron, and analogous results on d-dimensional simplex are derived by considering invariant and anti-invariant elements. Our main results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the simplex is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of (log n) ^d . The basic trigonometric functions on the simplex can be identified with Chebyshev polynomials in several variables already appeared in literature. We study common zeros of these polynomials and show that they are nodes for a family of Gaussian cubature formulas, which provides only the second known example of such formulas.     

Optimale Approximation mit Nebenbedingungen an lineare Funktionale auf periodischen Funktionen

Summary In a general Hilbert space of periodic functions numerical approximations with equidistant nodes for any bounded linear functional are given which are of minimal error norm in the class of approximations being exact for certain trigonometric polynomials. In examples optimal quadrature formulas with such side conditions are considered.     

The compass (star) identity for vector-valued rational interpolants

The compass identity (Wynn's five point star identity) for Padé approximants connects neighbouring elements called N, S, E, W and C in the Padé table. Its form has been extended to the cases of rational interpolation of ordinary (scalar) data and interpolation of vector-valued data. In this paper, full specifications of the associated five point identity for the scalar denominator polynomials and the vector numerator polynomials of the vector-valued rational interpolants on real data points are given, as well as the related generalisations of Frobenius' identities. Unique minimal forms of the polynomials constituting the interpolants and results about unattainable points correspond closely to their counterparts in the scalar case. This revised version was published online in June 2006 with corrections to the Cover Date.     

Sharp constant in Jackson’s inequality with modulus of smoothness for uniform approximations of periodic functions

It is proved that, in the space C_2π, for all k, n ∈ ?,n > 1, the following inequalities hold: where e _n?1(f) is the value of the best approximation of f by trigonometric polynomials and ω ₂(f, h) is the modulus of smoothness of f. A similar result is also obtained for approximation by continuous polygonal lines with equidistant nodes.     

Vector-valued,rational interpolants III

In 1963, Wynn proposed a method for rational interpolation of vector-valued quantities given on a set of distinct interpolation points. He used continued fractions, and generalized inverses for the reciprocals of vector-valued quantities. In this paper, we present an axiomatic approach to vector-valued rational interpolation. Uniquely defined interpolants are constructed for vector-valued data so that the components of the resulting vector-valued rational interpolant share a common denominator polynomial. An explicit determinantal formula is given for the denominator polynomial for the cases of (i) vector-valued rational interpolation on distinct real or complex points and (ii) vector-valued Padé approximation. We derive the connection with theε-algorithm of Wynn and Claessens, and we establish a five-term recurrence relation for the denominator polynomials.     

On polyharmonic interpolation

In the present paper we will introduce a new approach to multivariate interpolation by employing polyharmonic functions as interpolants, i.e. by solutions of higher order elliptic equations. We assume that the data arise from C^∞ or analytic functions in the ball BR. We prove two main results on the interpolation of C^∞ or analytic functions f in the ball BR by polyharmonic functions h of a given order of polyharmonicity p.     

The use of the stochastic arithmetic to estimate the value of interpolation polynomial with optimal degree

One of the considerable discussions in data interpolation is to find the optimal number of data which minimizes the error of the interpolation polynomial. In this paper, first the theorems corresponding to the equidistant nodes and the roots of the Chebyshev polynomials are proved in order to estimate the accuracy of the interpolation polynomial, when the number of data increases. Based on these theorems, then we show that by using a perturbation method based on the CESTAC method, it is possible to find the optimal degree of the interpolation polynomial. The results of numerical experiments are presented.     

Restricted even permutations and Chebyshev polynomials

We study generating functions for the number of even (odd) permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.     

OPTIMAL ERROR ESTIMATES OF THE PARTITION OF UNITY METHOD WITH LOCAL POLYNOMIAL APPROXIMATION SPACES

In this paper, we provide a theoretical analysis of the partition of unity finite elementmethod (PUFEM), which belongs to the family of meshfree methods. The usual erroranalysis only shows the order of error estimate to the same as the local approximations[12].Using standard linear finite element base functions as partition of unity and polynomials aslocal approximation space, in 1-d case, we derive optimal order error estimates for PUFEMinterpolants. Our analysis show that the error estimate is of one order higher than thelocal approximations. The interpolation error estimates yield optimal error estimates forPUFEM solutions of elliptic boundary value problems.     

Hermite interpolation with trigonometric polynomials

Interpolation methods of Hermite type in translation invariant spaces of trigonometric polynomials for any position of interpolation points and any number of derivatives are constructed. For the case of an odd number of interpolation conditions-periodic trigonometric polynomials of minimum order are chosen as interpolation functions while for the case of an even number of interpolation conditions-antiperiodic trigonometric polynomials of minimum order are appropriate.