Analysis of the combined finite element and Fourier interpolation

Summary We consider Lagrange interpolation involving trigonometric polynomials of degree N in one space direction, and piecewise polynomials over a finite element decomposition of mesh size h in the other space directions. We provide error estimates in non-isotropic Sobolev norms, depending additively on the parametersh andN. An application to the convergence analysis of an elliptic problem, with some numerical results, is given.     

Mean Convergence of Extended Lagrange Interpolation with Freud Weights

Let W := exp(-Q), where Q is of smooth polynomial growth at , for example Q(x) = |x|, > 1. We call W² a Freud weight. The mean convergence of Lagrange interpolation at the zeros of the orthonormal polynomials associated with the Freud weight WW² has been studied by several authors, as has the Lebesgue function of Lagrange interpolation. J. Szabados had the idea to add two additional points of interpolation, thereby reducing the Lebesgue constant to grow no faster than log n. In this paper, we show that mean convergence of Lagrange interpolation at this extended set of nodes displays a similar advantage over merely using the zeros of the orthogonal polynomials.     

On a generalization of compound Newton-Cotes quadrature formulas

We consider quadrature formulas defined by piecewise polynomial interpolation at equidistant nodes, admitting the nodes of adjacent polynomials to overlap, which generalizes the interpolation scheme of the compound Newton-Cotes quadrature formulas. The error constantse _,n in the estimate

改进Lagrange插值多项式误差上界的系数

当用Lagrange插值多项式逼近函数时,重要的是要了解误差项的性态.本文研究具有等距节点的Lagrange插值多项式,估计了Lagrange插值多项式逼近函数误差项的上界,改进了小于5次Lagrange插值多项式逼近函数误差界的系数.     

A bijection between Littlewood-Richardson tableaux and rigged configurations

We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials labeled by a partition and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of multiplicities of the irreducible -module of highest weight in the tensor product .     

Pointwise optimality of Bayesian wavelet estimators

We consider pointwise mean squared errors of several known Bayesian wavelet estimators, namely, posterior mean, posterior median and Bayes Factor, where the prior imposed on wavelet coefficients is a mixture of an atom of probability zero and a Gaussian density. We show that for the properly chosen hyperparameters of the prior, all the three estimators are (up to a log-factor) asymptotically minimax within any prescribed Besov ball . We discuss the Bayesian paradox and compare the results for the pointwise squared risk with those for the global mean squared error.     

Nonsymmetric interpolation macdonald polynomials and \mathfrak{g}\mathfrak{l}_n basic hypergeometric series

The Knop-Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polynomials to study a new type of basic hypergeometric series of type . Our main results include a new q-binomial theorem, a new q-Gauss sum, and several transformation formulae for series. *Supported by the ANR project MARS (BLAN06-2 134516). **Supported by the NSF grant DMS-0401387. ***Supported by the Australian Research Council.     

Smolyak's algorithm for weighted <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-approximation of multivariate functions with bounded <Emphasis Type="Italic">r</Emphasis>th mixed derivatives over ℝ<Superscript>d</Superscript>

We are interested in numerical algorithms for weighted L₁ approximation of functions defined on . We consider the space ℱ_r,d which consists of multivariate functions whose all mixed derivatives of order r are bounded in L₁-norm. We approximate f∈ℱ_r,d by an algorithm which uses evaluations of the function. The error is measured in the weighted L₁-norm with a weight function ρ. We construct and analyze Smolyak's algorithm for solving this problem. The algorithm is based on one-dimensional piecewise polynomial interpolation of degree at most r−1, where the interpolation points are specially chosen dependently on the smoothness parameter r and the weight ρ. We show that, under some condition on the rate of decay of ρ, the error of the proposed algorithm asymptotically behaves as , where n denotes the number of function evaluations used. The asymptotic constant is known and it decreases to zero exponentially fast as d→∞.     

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.     

The zeros of the Weierstrass $$\wp$$–function and hypergeometric series

We express the zeros of the Weierstrass -function in terms of generalized hypergeometric functions. As an application of our main result we prove the transcendence of two specific hypergeometric functions at algebraic arguments in the unit disc. We also give a Saalschützian ₄ F ₃–evaluation. Research of W. Duke was supported in part by NSF Grant DMS-0355564. He wishes to acknowledge and thank the Forschungsinstitut für Mathematik of ETH Zürich for its hospitality and support.     

Logarithmic moduli spaces for surfaces of class VII

In this paper we describe logarithmic moduli spaces of pairs (S, D) consisting of a minimal surface S of class VII with second Betti number b ₂ > 0 together with a reduced maximal divisor D of b ₂ rational curves. The special case of Enoki surfaces has already been considered by Dloussky and Kohler. We use normal forms for the action of the fundamental group of S\D and for the associated holomorphic contraction . Part of this work was done while the first author visited the University of Osnabrück under the program “Globale Methoden in der komplexen Geometrie” of the DFG and while the second author visited the Max-Planck-Institut für Mathematik in Bonn and the LATP, Université de Provence. We thank these institutions for their hospitality and for financial support. Furthermore the authors wish to thank Georges Dloussky for numerous discussions on surfaces of class VII.     

Kergin interpolants at the roots of unity approximate C2 functions

We establish a new formula for Kergin interpolation in the plane and use it to prove that the Kergin interpolation polynomials at the roots of unity of a function of classC ² in a neighborhood of the unit disc converge uniformly to the function on .     

Dedekind–Carlitz polynomials as lattice-point enumerators in rational polyhedra

We study higher-dimensional analogs of the Dedekind–Carlitz polynomials , where u and v are indeterminates and a and b are positive integers. Carlitz proved that these polynomials satisfy the reciprocity law from which one easily deduces many classical reciprocity theorems for the Dedekind sum and its generalizations. We illustrate that Dedekind–Carlitz polynomials appear naturally in generating functions of rational cones and use this fact to give geometric proofs of the Carlitz reciprocity law and various extensions of it. Our approach gives rise to new reciprocity theorems and computational complexity results for Dedekind–Carlitz polynomials, a characterization of Dedekind–Carlitz polynomials in terms of generating functions of lattice points in triangles, and a multivariate generalization of the Mordell–Pommersheim theorem on the appearance of Dedekind sums in Ehrhart polynomials of 3-dimensional lattice polytopes. Research of Haase supported by DFG Emmy Noether fellowship HA 4383/1. We thank Robin Chapman, Eric Mortenson, and an anonymous referee for helpful comments.     

A condition of Schoenberg-Whitney type for multivariate spline interpolation

Lagrange interpolation by finite-dimensional spaces of multivariate spline functions defined on a polyhedral regionK in ^k is studied. A condition of Schoenberg-Whitney type is introduced. The main result of this paper shows that this condition characterizes all configurationsT inK such that in every neighborhood ofT inK there must exist a configuration which admits unique Lagrange interpolation.     

Some Combinatorial and Analytical Identities

We give new proofs and explain the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, Uchimura, and Chen and Liu. We use the theory of basic hypergeometric functions, and generalize these identities. We also exploit the theory of polynomial expansions in the Wilson and Askey-Wilson bases to derive new identities which are not in the hierarchy of basic hypergeometric series. We demonstrate that a Lagrange interpolation formula always leads to very-well-poised basic hypergeometric series. As applications we prove that the Watson transformation of a balanced ${_{4}\phi_{3}}$ to a very-well-poised ${_{8}\phi_{7}}$ is equivalent to the Rodrigues-type formula for the Askey-Wilson polynomials. By applying the Leibniz formula for the Askey-Wilson operator we also establish the ${_{8}\phi_{7}}$ summation theorem.     

Selection of the number of regression variables; A minimax choice of generalized FPE

Summary A generalized Final Prediction Error (FPE_α)_ criterion is considered. Based onn observations, the numberk of regression variables is selected from a given range 0≦k≦K, so as to minimize . It is shown that if α tends to infinity withn, the selection is consistent but the maximum of the mean squared error of estimates of parameters diverges to infinity with the same order of divergence as that of α. A meaningful minimax choice of α exists for a regret type mean squared error, while for simple mean squared error it is trivially 0. The minimax regret choice of α converges to a constant, approximately 3.5 forK≧8 ifn−K increases simultaneously withn, otherwise it diverges to infinity withn.     

A path to the Arrow–Debreu competitive market equilibrium

We present polynomial-time interior-point algorithms for solving the Fisher and Arrow–Debreu competitive market equilibrium problems with linear utilities and n players. Both of them have the arithmetic operation complexity bound of )) for computing an -equilibrium solution. If the problem data are rational numbers and their bit-length is L, then the bound to generate an exact solution is O(n ⁴ L) which is in line with the best complexity bound for linear programming of the same dimension and size. This is a significant improvement over the previously best bound )) for approximating the two problems using other methods. The key ingredient to derive these results is to show that these problems admit convex optimization formulations, efficient barrier functions and fast rounding techniques. We also present a continuous path leading to the set of the Arrow–Debreu equilibrium, similar to the central path developed for linear programming interior-point methods. This path is derived from the weighted logarithmic utility and barrier functions and the Brouwer fixed-point theorem. The defining equations are bilinear and possess some primal-dual structure for the application of the Newton-based path-following method. Dedicated to Clovis Gonzaga on the occassion of his 60th birthday. This author was supported in part by NSF Grants DMS-0306611 and DMS-0604513. The author would like to thank Curtis Eaves, Osman Güler, Kamal Jain and Mike Todd for insightful discussions on this subject, especially on their mathematical references and economic interpretations of the fixed-point model presented in this paper.     

Hermite Subdivision Schemes and Taylor Polynomials

We propose a general study of the convergence of a Hermite subdivision scheme ℋ of degree d>0 in dimension 1. This is done by linking Hermite subdivision schemes and Taylor polynomials and by associating a so-called Taylor subdivision (vector) scheme . The main point of investigation is a spectral condition. If the subdivision scheme of the finite differences of is contractive, then is C ⁰ and ℋ is C ^d . We apply this result to two families of Hermite subdivision schemes. The first one is interpolatory; the second one is a kind of corner cutting. Both of them use the Tchakalov-Obreshkov interpolation polynomial.