32,693 research outputs found
Some classical multiple orthogonal polynomials
Recently there has been a renewed interest in an extension of the notion of
orthogonal polynomials known as multiple orthogonal polynomials. This notion
comes from simultaneous rational approximation (Hermite-Pade approximation) of
a system of several functions. We describe seven families of multiple
orthogonal polynomials which have he same flavor as the very classical
orthogonal polynomials of Jacobi, Laguerre and Hermite. We also mention some
open research problems and some applications
Modified Bernstein Polynomials and Jacobi Polynomials in q-Calculus
We introduce here a generalization of the modified Bernstein polynomials for
Jacobi weights using the -Bernstein basis proposed by G.M. Phillips to
generalize classical Bernstein Polynomials. The function is evaluated at points
which are in geometric progression in . Numerous properties of the
modified Bernstein Polynomials are extended to their -analogues:
simultaneous approximation, pointwise convergence even for unbounded functions,
shape-preserving property, Voronovskaya theorem, self-adjointness. Some
properties of the eigenvectors, which are -extensions of Jacobi polynomials,
are given
Polynomial Approximation in Sobolev Spaces on the Unit Sphere and the Unit Ball
This work is a continuation of the recent study by the authors on
approximation theory over the sphere and the ball. The main results define new
Sobolev spaces on these domains and study polynomial approximations for
functions in these spaces, including simultaneous approximation by polynomials
and relation between best approximation to a function and to its derivatives.Comment: 16 page
Nikishin systems are perfect
K. Mahler introduced the concept of perfect systems in the general theory he
developed for the simultaneous Hermite-Pade approximation of analytic
functions. We prove that Nikishin systems are perfect providing, by far, the
largest class of systems of functions for which this important property holds.
As consequences, in the context of Nikishin systems, we obtain: an extension of
Markov's theorem to simultaneous Hermite-Pade approximation, a general result
on the convergence of simultaneous quadrature rules of Gauss-Jacobi type, the
logarithmic asymptotics of general sequences of multiple orthogonal
polynomials, and an extension of the Denisov-Rakhmanov theorem for the ratio
asymptotics of mixed type multiple orthogonal polynomials.Comment: 39 page
Cauchy Biorthogonal Polynomials
The paper investigates the properties of certain biorthogonal polynomials
appearing in a specific simultaneous Hermite-Pade' approximation scheme.
Associated to any totally positive kernel and a pair of positive measures on
the positive axis we define biorthogonal polynomials and prove that their
zeroes are simple and positive. We then specialize the kernel to the Cauchy
kernel 1/{x+y} and show that the ensuing biorthogonal polynomials solve a
four-term recurrence relation, have relevant Christoffel-Darboux generalized
formulae, and their zeroes are interlaced. In addition, these polynomial solve
a combination of Hermite-Pade' approximation problems to a Nikishin system of
order 2. The motivation arises from two distant areas; on one side, in the
study of the inverse spectral problem for the peakon solution of the
Degasperis-Procesi equation; on the other side, from a random matrix model
involving two positive definite random Hermitian matrices. Finally, we show how
to characterize these polynomials in term of a Riemann-Hilbert problem.Comment: 38 pages, partially replaces arXiv:0711.408
- …