8,905 research outputs found
Bivariate Lagrange interpolation at the node points of Lissajous curves - the degenerate case
In this article, we study bivariate polynomial interpolation on the node
points of degenerate Lissajous figures. These node points form Chebyshev
lattices of rank and are generalizations of the well-known Padua points. We
show that these node points allow unique interpolation in appropriately defined
spaces of polynomials and give explicit formulas for the Lagrange basis
polynomials. Further, we prove mean and uniform convergence of the
interpolating schemes. For the uniform convergence the growth of the Lebesgue
constant has to be taken into consideration. It turns out that this growth is
of logarithmic nature.Comment: 26 pages, 6 figures, 1 tabl
On the Divergence Phenomenon in Hermite–Fejér Interpolation
AbstractGeneralizing results of L. Brutman and I. Gopengauz (1999, Constr. Approx.15, 611–617), we show that for any nonconstant entire function f and any interpolation scheme on [−1, 1], the associated Hermite–Fejér interpolating polynomials diverge on any infinite subset of C\[−1, 1]. Moreover, it turns out that even for the locally uniform convergence on the open interval ]−1, 1[ it is necessary that the interpolation scheme converges to the arcsine distribution
On the filtered polynomial interpolation at Chebyshev nodes
The paper deals with a special filtered approximation method, which
originates interpolation polynomials at Chebyshev zeros by using de la Vall\'ee
Poussin filters. These polynomials can be an useful device for many theoretical
and applicative problems since they combine the advantages of the classical
Lagrange interpolation, with the uniform convergence in spaces of locally
continuous functions equipped with suitable, Jacobi--weighted, uniform norms.
The uniform boundedness of the related Lebesgue constants, which equals to the
uniform convergence and is missing from Lagrange interpolation, has been
already proved in literature under different, but only sufficient, assumptions.
Here, we state the necessary and sufficient conditions to get it. These
conditions are easy to check since they are simple inequalities on the
exponents of the Jacobi weight defining the norm. Moreover, they are necessary
and sufficient to get filtered interpolating polynomials with a near best
approximation error, which tends to zero as the number of nodes tends to
infinity. In addition, the convergence rate is comparable with the error of
best polynomial approximation of degree , hence the approximation order
improves with the smoothness of the sought function. Several numerical
experiments are given in order to test the theoretical results, to make a
comparison with the Lagrange interpolation at the same nodes and to show how
the Gibbs phenomenon can be strongly reduced.Comment: 20 pages, 19 figures given in 8 eps file
Equidistribution of the Fekete points on the sphere
The Fekete points are the points that maximize a Vandermonde-type determinant
that appears in the polynomial Lagrange interpolation formula. They are well
suited points for interpolation formulas and numerical integration. We prove
the asymptotic equidistribution of the Fekete points in the sphere. The way we
proceed is by showing their connection with other array of points, the
Marcinkiewicz-Zygmund arrays and the interpolating arrays, that have been
studied recently
Equidistribution of the Fekete points on the sphere
The Fekete points are the points that maximize a Vandermonde-type determinant
that appears in the polynomial Lagrange interpolation formula. They are well
suited points for interpolation formulas and numerical integration. We prove
the asymptotic equidistribution of the Fekete points in the sphere. The way we
proceed is by showing their connection with other array of points, the
Marcinkiewicz-Zygmund arrays and the interpolating arrays, that have been
studied recently
Equivalence Theorems in Numerical Analysis : Integration, Differentiation and Interpolation
We show that if a numerical method is posed as a sequence of operators acting
on data and depending on a parameter, typically a measure of the size of
discretization, then consistency, convergence and stability can be related by a
Lax-Richtmyer type equivalence theorem -- a consistent method is convergent if
and only if it is stable. We define consistency as convergence on a dense
subspace and stability as discrete well-posedness. In some applications
convergence is harder to prove than consistency or stability since convergence
requires knowledge of the solution. An equivalence theorem can be useful in
such settings. We give concrete instances of equivalence theorems for
polynomial interpolation, numerical differentiation, numerical integration
using quadrature rules and Monte Carlo integration.Comment: 18 page
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