9,827 research outputs found
Zeros of linear combinations of Laguerre polynomials from different sequences
We study interlacing properties of the zeros of two types of linear
combinations of Laguerre polynomials with different parameters, namely
and .
Proofs and numerical counterexamples are given in situations where the zeros of
, and , respectively, interlace (or do not in general) with the zeros
of , , or . The results we prove hold
for continuous, as well as integral, shifts of the parameter
Multivariate Orthogonal Polynomials and Modified Moment Functionals
Multivariate orthogonal polynomials can be introduced by using a moment
functional defined on the linear space of polynomials in several variables with
real coefficients. We study the so-called Uvarov and Christoffel modifications
obtained by adding to the moment functional a finite set of mass points, or by
multiplying it times a polynomial of total degree 2, respectively. Orthogonal
polynomials associated with modified moment functionals will be studied, as
well as the impact of the modification in useful properties of the orthogonal
polynomials. Finally, some illustrative examples will be given
3nj-coefficients of su(1,1) as connection coefficients between orthogonal polynomials in n variables
Antisymmetric Orbit Functions
In the paper, properties of antisymmetric orbit functions are reviewed and
further developed. Antisymmetric orbit functions on the Euclidean space
are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a
Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such
functions are described. These functions are closely related to irreducible
characters of a compact semisimple Lie group of rank . Up to a sign,
values of antisymmetric orbit functions are repeated on copies of the
fundamental domain of the affine Weyl group (determined by the initial Weyl
group) in the entire Euclidean space . Antisymmetric orbit functions are
solutions of the corresponding Laplace equation in , vanishing on the
boundary of the fundamental domain . Antisymmetric orbit functions determine
a so-called antisymmetrized Fourier transform which is closely related to
expansions of central functions in characters of irreducible representations of
the group . They also determine a transform on a finite set of points of
(the discrete antisymmetric orbit function transform). Symmetric and
antisymmetric multivariate exponential, sine and cosine discrete transforms are
given.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
A Lanczos Method for Approximating Composite Functions
We seek to approximate a composite function h(x) = g(f(x)) with a global
polynomial. The standard approach chooses points x in the domain of f and
computes h(x) at each point, which requires an evaluation of f and an
evaluation of g. We present a Lanczos-based procedure that implicitly
approximates g with a polynomial of f. By constructing a quadrature rule for
the density function of f, we can approximate h(x) using many fewer evaluations
of g. The savings is particularly dramatic when g is much more expensive than f
or the dimension of x is large. We demonstrate this procedure with two
numerical examples: (i) an exponential function composed with a rational
function and (ii) a Navier-Stokes model of fluid flow with a scalar input
parameter that depends on multiple physical quantities
Onsager's algebra and partially orthogonal polynomials
The energy eigenvalues of the superintegrable chiral Potts model are
determined by the zeros of special polynomials which define finite
representations of Onsager's algebra. The polynomials determining the
low-sector eigenvalues have been given by Baxter in 1988. In the Z_3-case they
satisfy 4-term recursion relations and so cannot form orthogonal sequences.
However, we show that they are closely related to Jacobi polynomials and
satisfy a special "partial orthogonality" with respect to a Jacobi weight
function.Comment: 8 pages, no figure
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