2,567 research outputs found
Convexity of the zeros of some orthogonal polynomials and related functions
We study convexity properties of the zeros of some special functions that
follow from the convexity theorem of Sturm. We prove results on the intervals
of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials,
as well as functions related to them, using transformations under which the
zeros remain unchanged. We give upper as well as lower bounds for the distance
between consecutive zeros in several cases
Spacing properties of the zeros of orthogonal polynomials on Cantor sets via a sequence of polynomial mappings
Let be a probability measure with an infinite compact support on
. Let us further assume that is a sequence of
orthogonal polynomials for where is a sequence of
nonlinear polynomials and for all
. We prove that if there is an such that
is a root of for each then the distance between any two
zeros of an orthogonal polynomial for of a given degree greater than
has a lower bound in terms of the distance between the set of critical points
and the set of zeros of some . Using this, we find sharp bounds from below
and above for the infimum of distances between the consecutive zeros of
orthogonal polynomials for singular continuous measures.Comment: Contains less typo
Legendre-Gauss-Lobatto grids and associated nested dyadic grids
Legendre-Gauss-Lobatto (LGL) grids play a pivotal role in nodal spectral
methods for the numerical solution of partial differential equations. They not
only provide efficient high-order quadrature rules, but give also rise to norm
equivalences that could eventually lead to efficient preconditioning techniques
in high-order methods. Unfortunately, a serious obstruction to fully exploiting
the potential of such concepts is the fact that LGL grids of different degree
are not nested. This affects, on the one hand, the choice and analysis of
suitable auxiliary spaces, when applying the auxiliary space method as a
principal preconditioning paradigm, and, on the other hand, the efficient
solution of the auxiliary problems. As a central remedy, we consider certain
nested hierarchies of dyadic grids of locally comparable mesh size, that are in
a certain sense properly associated with the LGL grids. Their actual
suitability requires a subtle analysis of such grids which, in turn, relies on
a number of refined properties of LGL grids. The central objective of this
paper is to derive just these properties. This requires first revisiting
properties of close relatives to LGL grids which are subsequently used to
develop a refined analysis of LGL grids. These results allow us then to derive
the relevant properties of the associated dyadic grids.Comment: 35 pages, 7 figures, 2 tables, 2 algorithms; Keywords:
Legendre-Gauss-Lobatto grid, dyadic grid, graded grid, nested grid
Inequalities for Extreme Zeros of Some Classical Orthogonal and q-orthogonal Polynomials
Let {pn}1 n=0 be a sequence of orthogonal polynomials. We briefly review properties of pn that have been used to derive upper and lower bounds for the largest and smallest zero of pn. Bounds for the extreme zeros of Laguerre, Jacobi and Gegenbauer polynomials that have been obtained using different approaches are numerically compared and new bounds for extreme zeros of q-Laguerre and little q-Jacobi polynomials are proved
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