6,499 research outputs found
On discrete orthogonal polynomials of several variables
Let be a set of isolated points in \RR^d. Define a linear functional
\CL on the space of real polynomials restricted on , \CL f = \sum_{x \in
V} f(x)\rho(x), where is a nonzero function on . Polynomial
subspaces that contain discrete orthogonal polynomials with respect to the
bilinear form = \CL(f g) are identified. One result shows that the
discrete orthogonal polynomials still satisfy a three-term relation and
Favard's theorem holds in this general setting.Comment: 15 pages, 2 figure
Moving least squares via orthogonal polynomials
A method for moving least squares interpolation and differentiation is
presented in the framework of orthogonal polynomials on discrete points. This
yields a robust and efficient method which can avoid singularities and
breakdowns in the moving least squares method caused by particular
configurations of nodes in the system. The method is tested by applying it to
the estimation of first and second derivatives of test functions on random
point distributions in two and three dimensions and by examining in detail the
evaluation of second derivatives on one selected configuration. The accuracy
and convergence of the method are examined with respect to length scale (point
separation) and the number of points used. The method is found to be robust,
accurate and convergent.Comment: Extensively revised in response to referees' comment
Fast algorithm for border bases of Artinian Gorenstein algebras
Given a multi-index sequence , we present a new efficient algorithm
to compute generators of the linear recurrence relations between the terms of
. We transform this problem into an algebraic one, by identifying
multi-index sequences, multivariate formal power series and linear functionals
on the ring of multivariate polynomials. In this setting, the recurrence
relations are the elements of the kerne l\sigma of the Hankel operator
$H$\sigma associated to . We describe the correspondence between
multi-index sequences with a Hankel operator of finite rank and Artinian
Gorenstein Algebras. We show how the algebraic structure of the Artinian
Gorenstein algebra \sigma\sigma yields the
structure of the terms $\sigma\alpha N nAK[x 1 ,. .. , xnIHIA$ and the tables of multiplication by the variables in these
bases. It is an extension of Berlekamp-Massey-Sakata (BMS) algorithm, with
improved complexity bounds. We present applications of the method to different
problems such as the decomposition of functions into weighted sums of
exponential functions, sparse interpolation, fast decoding of algebraic codes,
computing the vanishing ideal of points, and tensor decomposition. Some
benchmarks illustrate the practical behavior of the algorithm
Rodrigues Formula for Hi-Jack Symmetric Polynomials Associated with the Quantum Calogero Model
The Hi-Jack symmetric polynomials, which are associated with the simultaneous
eigenstates for the first and second conserved operators of the quantum
Calogero model, are studied. Using the algebraic properties of the Dunkl
operators for the model, we derive the Rodrigues formula for the Hi-Jack
symmetric polynomials. Some properties of the Hi-Jack polynomials and the
relationships with the Jack symmetric polynomials and with the basis given by
the QISM approach are presented. The Hi-Jack symmetric polynomials are strong
candidates for the orthogonal basis of the quantum Calogero model.Comment: 17 pages, LaTeX file using jpsj.sty (ver. 0.8), cite.sty,
subeqna.sty, subeqn.sty, jpsjbs1.sty and jpsjbs2.sty (all included.) You can
get all the macros from ftp.u-tokyo.ac.jp/pub/SOCIETY/JPSJ
Properties of some families of hypergeometric orthogonal polynomials in several variables
Limiting cases are studied of the Koornwinder-Macdonald multivariable
generalization of the Askey-Wilson polynomials. We recover recently and not so
recently introduced families of hypergeometric orthogonal polynomials in
several variables consisting of multivariable Wilson, continuous Hahn and
Jacobi type polynomials, respectively. For each class of polynomials we provide
systems of difference (or differential) equations, recurrence relations, and
expressions for the norms of the polynomials in terms of the norm of the
constant polynomial.Comment: 42 pages, AMSLaTeX 1.1 with amssym
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