2,255 research outputs found
Chebyshev model arithmetic for factorable functions
This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden
Orthogonal polynomials of compact simple Lie groups
Recursive algebraic construction of two infinite families of polynomials in
variables is proposed as a uniform method applicable to every semisimple
Lie group of rank . Its result recognizes Chebyshev polynomials of the first
and second kind as the special case of the simple group of type . The
obtained not Laurent-type polynomials are proved to be equivalent to the
partial cases of the Macdonald symmetric polynomials. Basic relation between
the polynomials and their properties follow from the corresponding properties
of the orbit functions, namely the orthogonality and discretization. Recurrence
relations are shown for the Lie groups of types , , , ,
, , and together with lowest polynomials.Comment: 34 pages, some minor changes were done, to appear in IJMM
The finite Fourier transform of classical polynomials
The finite Fourier transform of a family of orthogonal polynomials
, is the usual transform of the polynomial extended by outside
their natural domain. Explicit expressions are given for the Legendre, Jacobi,
Gegenbauer and Chebyshev families
A momentum-space Argonne V18 interaction
This paper gives a momentum-space representation of the Argonne V18 potential
as an expansion in products of spin-isospin operators with scalar coefficient
functions of the momentum transfer. Two representations of the scalar
coefficient functions for the strong part of the interaction are given. One is
as an expansion in an orthonormal basis of rational functions and the other as
an expansion in Chebyshev polynomials on different intervals. Both provide
practical and efficient representations for computing the momentum-space
potential that do not require integration or interpolation. Programs based on
both expansions are available as supplementary material. Analytic expressions
are given for the scalar coefficient functions of the Fourier transform of the
electromagnetic part of the Argonne V18. A simple method for computing the
partial-wave projections of these interactions from the operator expressions is
also given.Comment: 61 pages. 26 figure
Chebyshev type lattice path weight polynomials by a constant term method
We prove a constant term theorem which is useful for finding weight
polynomials for Ballot/Motzkin paths in a strip with a fixed number of
arbitrary `decorated' weights as well as an arbitrary `background' weight. Our
CT theorem, like Viennot's lattice path theorem from which it is derived
primarily by a change of variable lemma, is expressed in terms of orthogonal
polynomials which in our applications of interest often turn out to be
non-classical. Hence we also present an efficient method for finding explicit
closed form polynomial expressions for these non-classical orthogonal
polynomials. Our method for finding the closed form polynomial expressions
relies on simple combinatorial manipulations of Viennot's diagrammatic
representation for orthogonal polynomials. In the course of the paper we also
provide a new proof of Viennot's original orthogonal polynomial lattice path
theorem. The new proof is of interest because it uses diagonalization of the
transfer matrix, but gets around difficulties that have arisen in past attempts
to use this approach. In particular we show how to sum over a set of implicitly
defined zeros of a given orthogonal polynomial, either by using properties of
residues or by using partial fractions. We conclude by applying the method to
two lattice path problems important in the study of polymer physics as models
of steric stabilization and sensitized flocculation.Comment: 27 pages, 14 figure
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