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1.
Summary.
With denoting the -th partial
sum of ${\rm e}^{z}$, the exact rate of convergence of the zeros of the
normalized partial sums, , to the Szeg\"o curve
was
recently studied by Carpenter et al. (1991), where
is defined by
Here, the above results are generalized to the convergence of
the zeros and poles of certain sequences of normalized Pad\'{e}
approximants
to , where is the associated Pad\'{e} rational approximation to .
Received February 2, 1994 相似文献
2.
3.
Summary. A nonstationary multiresolution of is generated by a sequence of scaling functions We consider that is the solution of the nonstationary refinement equations
where is finitely supported for each k and M is a dilation matrix. We study various forms of convergence in of the corresponding nonstationary cascade algorithm as k or n tends to It is assumed that there is a stationary refinement equation at with filter sequence h and that The results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of
the transition operator associated with h.
Received September 19, 1997 / Revised version received May 22, 1998 / Published online August 19, 1999 相似文献
4.
Convergence of multidimensional cascade algorithm 总被引:12,自引:0,他引:12
A necessary and sufficient condition on the spectrum of the restricted transition operator is given for the convergence in
of the multidimensional cascade algorithm for any starting function whose shifts form a partition of unity.
Received September 12, 1995 / Revised version received August 2, 1996 相似文献
5.
Summary.
We describe algorithms for constructing point sets at which interpolation by
spaces of bivariate splines of arbitrary degree and smoothness is
possible. The splines are defined on rectangular partitions adding
one or two diagonals to each rectangle. The interpolation sets
are selected in such a way that the grid points of the partition
are contained in these sets, and no large linear systems have to be solved.
Our method is to generate a net of line segments and to choose point sets in
these segments which satisfy the Schoenberg-Whitney condition for
certain univariate spline spaces such that a principle of degree
reduction can be applied. In order to include the grid points in the
interpolation sets, we give a sufficient Schoenberg-Whitney type
condition for interpolation by bivariate splines supported in certain cones.
This approach is completely different
from the known interpolation methods for bivariate splines of degree at most
three. Our method is illustrated by some numerical examples.
Received
October 5, 1992 / Revised version received May 13, 1994 相似文献
6.
Yuan Xu 《Numerische Mathematik》2000,85(1):155-173
Summary. We examine the method of reproducing kernel for constructing cubature formulae on the unit ball and on the triangle in light
of the compact formulae of the reproducing kernels that are discovered recently. Several new cubature formulae are derived.
Received April 15, 1998 / Revised version received November 24, 1998 / Published online January 27, 2000 相似文献
7.
Summary.
This paper presents a method to recover
exponential accuracy at all points (including at the
discontinuities themselves), from the knowledge
of an approximation to the
interpolation polynomial (or trigonometrical polynomial).
We show that if we are given the collocation point values
(or a highly accurate approximation) at the Gauss
or Gauss-Lobatto points,
we can reconstruct an uniform exponentially convergent
approximation to the function in any sub-interval
of analyticity. The proof covers the cases of Fourier,
Chebyshev, Legendre, and more
general Gegenbauer collocation methods.
A numerical example is also provided.
Received
July 17, 1994 / Revised version received December 12, 1994 相似文献
8.
Given a finite subset
of an additive group
such as
or
, we are interested in efficient covering of
by translates of
, and efficient packing of translates of
in
. A set
provides a covering if the translates
with
cover
(i.e., their union is
), and the covering will be efficient if
has small density in
. On the other hand, a set
will provide a packing if the translated sets
with
are mutually disjoint, and the packing is efficient if
has large density.
In the present part (I) we will derive some facts on these concepts when
, and give estimates for the minimal covering densities and maximal packing densities of finite sets
. In part (II) we will again deal with
, and study the behaviour of such densities under linear transformations. In part (III) we will turn to
.
Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395,
USA
The first author was partially supported by NSF DMS 0074531. 相似文献
9.
10.
11.
Michael J. Johnson 《Numerische Mathematik》2000,84(3):451-474
Summary. We show that the -norm of the error in thin-plate spline interpolation in the unit disc decays like , where , under the assumptions that the function to be approximated is and that the interpolation points contain the finite grid .
Received February 13, 1998 / Published online September 24, 1999 相似文献
12.
Summary. We generalize earlier results concerning
an asymptotic error expansion of wavelet
approximations. The properties of the monowavelets,
which are the building
blocks for the error expansion, are studied in more
detail, and connections
between spline wavelets and Euler and
Bernoulli polynomials are pointed out.
The expansion is used to compare the
error for different wavelet families.
We prove that the leading terms of the
expansion only depend on the multiresolution
subspaces and not
on how the complementary subspaces
are chosen.
Consequently, for a fixed set of
subspaces , the leading
terms do not depend on the fact whether
the wavelets are orthogonal or not.
We also show that Daubechies' orthogonal wavelets need,
in general, one level more than spline wavelets to obtain an
approximation with a prescribed accuracy.
These results are illustrated with numerical examples.
Received May 3, 1993 / Revised version received January 31, 1994 相似文献
13.
Thomas Sauer 《Numerische Mathematik》1997,78(1):59-85
Summary. Minimal degree interpolation spaces with respect to a finite set of points are subspaces of multivariate polynomials of least
possible degree for which Lagrange interpolation with respect to the given points is uniquely solvable and degree reducing.
This is a generalization of the concept of least interpolation introduced by de Boor and Ron. This paper investigates the
behavior of Lagrange interpolation with respect to these spaces, giving a Newton interpolation method and a remainder formula
for the error of interpolation. Moreover, a special minimal degree interpolation space will be introduced which is particularly
beneficial from the numerical point of view.
Received June 9, 1995 / Revised version received June 26, 1996 相似文献
14.
Motivated by earlier considerations of interval interpolation problems as well as a particular application to the reconstruction
of railway bridges, we deal with the problem of univariate convexity preserving interval interpolation. To allow convex interpolation,
the given data intervals have to be in (strictly) convex position. This property is checked by applying an abstract three-term
staircase algorithm, which is presented in this paper. Additionally, the algorithm provides strictly convex ordinates belonging
to the data intervals. Therefore, the known methods in convex Lagrange interpolation can be used to obtain interval interpolants.
In particular, we refer to methods based on polynomial splines defined on grids with additional knots.
Received September 22, 1997 / Revised version received May 26, 1998 相似文献
15.
16.
For arbitrary finite group and countable Dedekind domain such that the residue field is finite for every maximal -ideal , we show that the localizations at every maximal ideal of two -lattices are isomorphic if and only if the two lattices satisfy the same first order sentences. Then we investigate generalizations
of the above results to arbitrary -torsion-free -modules and we apply the previous results to show the decidability of the theory of -lattices. Eventually, we show that -lattices have undecidable theory.
Received November 28, 1995 相似文献
17.
Summary.
We consider convex interpolation with
cubic splines on
grids built by adding two knots in each subinterval of neighbouring data
sites. The additional knots have to be variable in order to get a chance
to always retain convexity. By means of the staircase algorithm we
provide computable intervals for the added knots such that all knots
from these intervals allow convexity preserving spline interpolation of
continuity.
Received
May 31, 1994 / Revised version received December 22, 1994 相似文献
18.
Shayne Waldron 《Numerische Mathematik》2000,85(3):469-484
Summary. Distribution theory is used to construct minimally supported Peano kernel type representations for linear functionals such as the error in multivariate Hermite interpolation. The simplest
case is that of representing the error in approximation to f by the constant polynomial f(a) in terms of integrals of the first order derivatives of f. This is discussed in detail. Here it is shown that suprisingly there exist many representations which are not minimally
supported, and involve the integration of first order derivatives over multidimensional regions. The distance of smooth functions
from the constants in the uniform norm is estimated using our representations for the error.
Received June 30, 1997 / Revised version received April 6, 1999 / Published online February 17, 2000 相似文献
19.
M. Kappert 《Numerische Mathematik》1996,74(4):397-417
Summary. Let denote the -th partial sum of the exponential function. Carpenter et al. (1991) [1] studied the exact rate of convergence of the zeros
of the normalized partial sums to the so-called Szeg?-curve Here we apply parts of the results found by Carpenter et al. to the zeros of the normalized partial sums of and .
Received August 11, 1995 相似文献
20.
Summary. We study here in detail the location of the real and complex zeros of the partial sums of and , which extends results of Szeg? (1924) and Kappert (1996).
Received November 9, 2000 / Published online August 17, 2001 相似文献