Lagrange interpolation on extended generalized Jacobi nodes

We consider theL ^p -convergence of interpolatory processes for nonsmooth functions. Therefore we use generalizations of the well-known Marcinkiewicz-Zygmund inequality for trigonometric polynomials to the case of algebraic polynomials, extending a result of Y. Xu. Particularly, we obtain the order of convergence for certain Lagrange and quasi-Lagrange interpolatory processes on generalized Jacobi nodes. Our approach enables us also to discuss the influence of additional nodes near the endpoints ±1.     

Some researches on multivariate Lagrange interpolation along the sufficiently intersected algebraic manifold

In this article, Lagrange interpolation by polynomials in several variables is studied. Particularly on the sufficiently intersected algebraic manifolds, we discuss the dimension about the interpolation space of polynomials. After defining properly posed set of nodes (or PPSN for short) along the sufficiently intersected algebraic manifolds, we prove the existence of PPSN and give the number of points in PPSN of any degree. Moreover, in order to compute the number of points in PPSN concretely, we propose the operator ? _k with reciprocal difference.     

Expansions for the Fundamental Hermite Interpolation Polynomials in Terms of Chebyshev Polynomials

We obtain explicit expansions of the fundamental Hermite interpolation polynomials in terms of Chebyshev polynomials in the case where the nodes considered are either zeros of the (n + 1)th-degree Chebyshev polynomial or extremum points of the nth-degree Chebyshev polynomial.     

The Divergence of Lagrange Interpolation for |x| at Equidistant Nodes

In 1918 S. N. Bernstein published the surprising result that the sequence of Lagrange interpolation polynomials to |x| at equally spaced nodes in [−1, 1] diverges everywhere, except at zero and the end-points. In the present paper, we prove that the sequence of Lagrange interpolation polynomials corresponding to |x|^α (0<α1) on equidistant nodes in [−1, 1] diverges everywhere in the interval except at zero and the end-points.     

A note on fast Fourier transforms for nonequispaced grids

In this paper, we are concerned with fast Fourier transforms for nonequispaced grids. We propose a general efficient method for the fast evaluation of trigonometric polynomials at nonequispaced nodes based on the approximation of the polynomials by special linear combinations of translates of suitable functions ϕ. We derive estimates for the approximation error. In particular, we improve the estimates given by Dutt and Rokhlin [7]. As a practical consequence, we obtain a criterion for the choice of the parameters involved in the fast transforms. This revised version was published online in August 2006 with corrections to the Cover Date.     

A limiting relation between Chebyschev polynomials with a discrete argument and Legendre polynomials

Asymptotic estimates are obtained in a uniform metric and in the L _p metrics (p 2) for the difference between Chebyschev polynomials with a discrete argument and Legendre polynomials, under simultaneous passage to infinity of the degree of the polynomials and the number of lattice nodes at which the Chebyschev polynomials are defined.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 37–43, 1989.     

On rational approximations of functions

Rational analogues of Taylor and Fourier polynomials and polynomials of the Lagrange type are constructed and investigated. These analogues are shown to approximate a function with a remainder term of the same order as in the case of the aforementioned polynomials. Conditions are established under which a polynomial of the Lagrange type and its rational analogue are two-sided approximations of a function on a segment and their derivatives are two-sided approximations of the derivative of the function at collocation nodes. Bibliography: 2 titles. Translated fromObchuslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 32–38.     

Functions approximated by any sequence of interpolation polynomials

In this note, we seek for functions f which are approximated by the sequence of interpolation polynomials of f obtained by any prescribed system of nodes.     

Computation of the differentiation matrix for the Hermite interpolating polynomials

We obtain formulas for computing the elements of the differentiation matrix for special cases of the Hermite interpolating polynomials. They are expressed in terms of the elements of the differentiation matrices of the Lagrange interpolating polynomials in various systems of interpolation nodes, which can easily be calculated on a computer. These formulas find application in numerical realization of collocation finiteelement methods for solving differential problems.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 43–49.     

A bivariate Markov inequality for Chebyshev polynomials of the second kind

This note presents a Markov-type inequality for polynomials in two variables where the Chebyshev polynomials of the second kind in either one of the variables are extremal. We assume a bound on a polynomial at the set of even or odd Chebyshev nodes with the boundary nodes omitted and obtain bounds on its even or odd order directional derivatives in a critical direction. Previously, the author has given a corresponding inequality for Chebyshev polynomials of the first kind and has obtained the extension of V.A. Markov’s theorem to real normed linear spaces as an easy corollary.To prove our inequality we construct Lagrange polynomials for the new class of nodes we consider and give a corresponding Christoffel–Darboux formula. It is enough to determine the sign of the directional derivatives of the Lagrange polynomials.     

Numerical integration with polynomial exactness over a spherical cap

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere \mathbbS²\mathbb{S}^2, we discuss tensor product rules with n ²/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n ³) nodes for numerical integration over spherical caps on \mathbbS²\mathbb{S}^2. For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that have O(n ^d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that are exact for all spherical polynomials of degree ≤ n have at least O(n ^d ) nodes and possess a certain regularity property.     

Sampling Sets and Quadrature Formulae on the Rotation Group

In this paper, we construct sampling sets over the rotation group SO(3). The proposed construction is based on a parameterization, which reflects the product nature ² × ¹ of SO(3) very well, and leads to a spherical Pythagorean-like formula in the parameter domain. We prove that by using uniformly distributed points on ² and ¹, we obtain uniformly sampling nodes on the rotation group SO(3). Furthermore, quadrature formulae on ² and ¹ lead to quadratures on SO(3), as well. For scattered data on SO(3), we give a necessary condition on the mesh norm such that the sampling nodes possess nonnegative quadrature weights. We propose an algorithm for computing the quadrature weights for scattered data on SO(3) based on fast algorithms. We confirm our theoretical results with examples and numerical tests.     

On Lagrange interpolation for |<Emphasis Type="Italic">X</Emphasis>|<Superscript><Emphasis Type="Italic">α</Emphasis></Superscript> (0 < <Emphasis Type="Italic">α</Emphasis> < 1)

We study the optimal order of approximation for |x| ^α (0 < α < 1) by Lagrange interpolation polynomials based on Chebyshev nodes of the first kind. It is proved that the Jackson order of approximation is attained. Supported by the NSFC, 10601065.     

On the Connection of Uncertainty Principles for Functions on the Circle and on the Real Line

An uncertainty principle for 2-periodic functions and the classical Heisenberg uncertainty principle are shown to be linked by a limit process. Dependent on a parameter, a function on the real line generates periodic functions either by periodization or sampling. It is proven that under certain smoothness conditions, the periodic uncertainty products of the generated functions converge to the real-line uncertainty product of the original function if the parameter tends to infinity. These results are used to find asymptotically optimal sequences for the periodic uncertainty principle, based either on Theta functions or trigonometric polynomials obtained by sampling B-splines.     

Error estimates for quadrature formulas

We construct two-sided polynomials of collocation type of the same order as a given system of basis functions according to a given ordered system of nodes of arbitrary multiplicity and according to a system of nodes displaced to the right (or to the left) at one position. Numerical estimates are given for the remaining terms of the quadrature formulas.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 70, pp. 21–31, 1990.     

On Gauss-Type Quadrature Rules

Some Gauss-type Quadrature rules over [0, 1], which involve values and/or the derivative of the integrand at 0 and/or 1, are investigated. Our work is based on the orthogonal polynomials with respect to linear weight function ω(t): = 1 ? t over [0, 1]. These polynomials are also linked with a class of recently developed “identity-type functions”. Along the lines of Golub's work, the nodes and weights of the quadrature rules are computed from Jacobi-type matrices with simple rational entries. Computational procedures for the derived rules are tested on different integrands. The proposed methods have some advantage over the respective Gauss-type rules with respect to the Gauss weight function ω(t): = 1 over [0, 1].     

Product Gauss cubature over polygons based on Green’s integration formula

We have implemented in Matlab a Gauss-like cubature formula over convex, nonconvex or even multiply connected polygons. The formula is exact for polynomials of degree at most 2n-1 using N∼mn ² nodes, m being the number of sides that are not orthogonal to a given line, and not lying on it. It does not need any preprocessing like triangulation of the domain, but relies directly on univariate Gauss–Legendre quadrature via Green’s integral formula. Several numerical tests are presented. AMS subject classification (2000)  65F20     

Analysis of an algorithm for generating locally optimal meshes for approximation by discontinuous piecewise polynomials

This paper discusses the problem of constructing a locally optimal mesh for the best approximation of a given function by discontinuous piecewise polynomials. In the one-dimensional case, it is shown that, under certain assumptions on the approximated function, Baines' algorithm [M.J. Baines, Math. Comp., 62 (1994), pp. 645-669] for piecewise linear or piecewise constant polynomials produces a mesh sequence which converges to an optimal mesh. The rate of convergence is investigated. A two-dimensional modification of this algorithm is proposed in which both the nodes and the connection between the nodes are self-adjusting. Numerical results in one and two dimensions are presented.

     

Spline approximations inL p on an interval

We consider approximations in the spaceL _p[0,a] to differentiable functions whoselth derivative belongs toL _p[0,a]. The function to be approximated is extended to the entire axis by Lagrange interpolation polynomials, and spline approximation with equally spaced nodes on the entire axis is then applied. This procedure results in a good approximation to the original function.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 281–294, August, 1995.The author is grateful to Yu. N. Subbotin for posing the problem and for his attention to the work, as well as to N. I. Chernykh for useful remarks.     

On Lagrange interpolatory parabolas to $\left | x\right | ^{\alpha }$ at equally spaced nodes

In this paper we present a generalized quantitative version of a result due to D. L. Berman concerning the exact convergence rate at zero of Lagrange interpolation polynomials to | x| ^a\left | x\right | ^{\alpha } based on equally spaced nodes in [-1, 1]. The estimates obtained turn out to be best possible.