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1.
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
2.
Summary. The cascade algorithm with mask a and dilation M generates a sequence by the iterative process
from a starting function where M is a dilation matrix. A complete characterization is given for the strong convergence of cascade algorithms in Sobolev spaces
for the case in which M is isotropic. The results on the convergence of cascade algorithms are used to deduce simple conditions for the computation
of integrals of products of derivatives of refinable functions and wavelets.
Received May 5, 1999 / Revised version received June 24, 1999 / Published online June 20, 2001 相似文献
3.
Summary. In this paper we combine an earlier method developed with K. Jetter on general cardinal interpolation with constructions
of compactly supported solutions for cardinal interpolation to gain compactly supported fundamental solutions for the general
interpolation problem. The general interpolation problem admits the interpolation of the functional and derivative values
under very weak restrictions on the derivatives to be interpolated. In the univariate case, some known general constructions
of compactly supported fundamental solutions for cardinal interpolation are discussed together with algorithms for their construction
that make use of MAPLE. Another construction based on finite decomposition and reconstruction for spline spaces is also provided.
Ideas used in the latter construction are lifted to provide a general construction of compactly supported fundamental solutions
for cardinal interpolation in the multivariate case. Examples are provided, several in the context of some general interpolation
problem to illustrate how easy is the transition from cardinal interpolation to general interpolation.
Received May 11, 1993 / Revised version received August 16, 1994 相似文献
4.
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
5.
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
6.
Summary.
We present generalizations of the nonsymmetric Levinson and Schur algorithms
for non-Hermitian Toeplitz matrices with some singular or
ill-conditioned
leading principal submatrices. The underlying recurrences allow us to
go from any pair of successive well-conditioned leading principal submatrices
to any such pair of larger order. If the look-ahead step size between
these pairs is bounded, our generalized Levinson and Schur recurrences
require $ operations, and the Schur recurrences can be combined
with recursive doubling so that an $ algorithm results.
The overhead (in operations and storage) of look-ahead steps is very small.
There are various options for applying these algorithms to solving linear
systems with Toeplitz matrix.
Received July 26, 1993 相似文献
7.
Juan José Torrens 《Numerische Mathematik》1997,76(1):69-85
Résumé. On établit des majorations de l'erreur d'approximation par éléments finis à partir de données de Lagrange pour des fonctions
appartenant à un espace de Sobolev d'ordre convenable, lorsque les degrés de liberté sont approchés à l'aide de la méthode
des plaquettes splines introduite par A. Le Méhauté (cf. [13], [14], [15]). Les résultats obtenus s'appliquent notamment
à la construction de surfaces de classe .
Received May 29, 1995 / Revised version received August 20, 1995 相似文献
8.
9.
Yuan Xu 《Numerische Mathematik》1994,69(2):233-241
Summary.
The existence of Gaussian cubature for a given measure
depends on whether the corresponding multivariate orthogonal polynomials have
enough common zeros. We examine a class of orthogonal
polynomials of two variables generated from that of one variable.
Received February 9, 1993 / Revised version received
January 18, 1994 相似文献
10.
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
11.
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
12.
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
13.
N. Papamichael I.E. Pritsker E.B. Saff N.S. Stylianopoulos 《Numerische Mathematik》1997,76(4):489-513
Summary. In this paper we examine the convergence rates in an adaptive version of an orthonormalization method for approximating the
conformal mapping of an annular region onto a circular annulus. In particular, we consider the case where has an analytic extension in compl() and, for this case, we determine optimal ray sequences of approximants that give the best possible geometric rate of uniform
convergence. We also estimate the rate of uniform convergence in the case where the annular region has piecewise analytic boundary without cusps. In both cases we also give the corresponding rates for the approximations
to the conformal module of .
Received February 2, 1996 相似文献
14.
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
15.
Summary.
Numerical methods are considered for generating polynomials
orthogonal with respect to an inner product of Sobolev type, i.e.,
one that involves derivatives up to some given order, each
having its own (positive) measure associated with it. The principal
objective is to compute the coefficients in the increasing-order
recurrence relation that these polynomials satisfy by virtue of
them forming a sequence of monic polynomials with degrees increasing
by 1 from one member to the next. As a by-product of this computation,
one gains access to the zeros of these polynomials via eigenvalues of
an upper Hessenberg matrix formed by the coefficients generated. Two
methods are developed: One is based on the modified moments of the
constitutive measures and generalizes what for ordinary orthogonal
polynomials is known as "modified Chebyshev algorithm". The
other - a generalization of "Stieltjes's procedure" -
expresses the desired coefficients in terms of a Sobolev inner product
involving the orthogonal polynomials in question, whereby the inner
product is evaluated by numerical quadrature and the polynomials
involved are computed by means of the recurrence relation already
generated up to that point. The numerical characteristics of these
methods are illustrated in the case of Sobolev orthogonal polynomials
of old as well as new types. Based on extensive numerical
experimentation, a number of conjectures are formulated with regard
to the location and interlacing properties of the respective zeros.
Received July 13, 1994 /
Revised version received September 26, 1994 相似文献
16.
K.A. Ariyawansa 《Numerische Mathematik》1998,80(3):363-376
Summary. Many successful quasi-Newton methods for optimization are based on positive definite local quadratic approximations to the
objective function that interpolate the values of the gradient at the current and new iterates. Line search termination criteria
used in such quasi-Newton methods usually possess two important properties. First, they guarantee the existence of such a
local quadratic approximation. Second, under suitable conditions, they allow one to prove that the limit of the component
of the gradient in the normalized search direction is zero. This is usually an intermediate result in proving convergence.
Collinear scaling algorithms proposed initially by Davidon in 1980 are natural extensions of quasi-Newton methods in the sense
that they are based on normal conic local approximations that extend positive definite local quadratic approximations, and
that they interpolate values of both the gradient and the function at the current and new iterates. Line search termination criteria that guarantee the existence
of such a normal conic local approximation, which also allow one to prove that the component of the gradient in the normalized
search direction tends to zero, are not known. In this paper, we propose such line search termination criteria for an important
special case where the function being minimized belongs to a certain class of convex functions.
Received February 1, 1997 / Revised version received September 8, 1997 相似文献
17.
Roland W. Freund 《Numerische Mathematik》1994,68(1):35-69
Summary.
The Bareiss algorithm is one of the classical fast
solvers for systems of linear equations with Toeplitz
coefficient matrices.
The method takes advantage of the special structure,
and it computes the solution of a Toeplitz system
of order~ with only~
arithmetic operations, instead of~
operations.
However, the original Bareiss algorithm requires that
all leading principal submatrices be nonsingular, and
the algorithm is numerically unstable if singular or ill-conditioned
submatrices occur.
In this paper, an extension of the Bareiss algorithm to
general Toeplitz systems is presented.
Using look-ahead techniques, the proposed algorithm can skip over
arbitrary blocks of singular or ill-conditioned submatrices,
and at the same time, it still fully exploits the Toeplitz
structure.
Implementation details and operations counts are given, and
numerical experiments are reported.
We also discuss special versions of the proposed look-ahead Bareiss
algorithm for Hermitian indefinite Toeplitz systems and
banded Toeplitz systems.
Received August 27, 1993 / Revised version received March 1994 相似文献
18.
Summary. Macro-elements of smoothness on Clough-Tocher triangle splits are constructed for all . These new elements are improvements on elements constructed in [11] in that (disproving a conjecture made there) certain
unneeded degrees of freedom have been removed. Numerical experiments on Hermite interpolation with the new elements are included.
Received September 6, 2000 / Revised version received November 15, 2000 / Published online July 25, 2001 相似文献
19.
Summary. The objective of this paper is to introduce a fast algorithm for computing the integral wavelet transform (IWT) on a dense
set of points in the time-scale domain. By applying the duality principle and using a compactly supported spline-wavelet as
the analyzing wavelet, this fast integral wavelet transform (FIWT) is realized by applying only FIR (moving average) operations,
and can be implemented in parallel. Since this computational procedure is based on a local optimal-order spline interpolation
scheme and the FIR filters are exact, the IWT values so obtained are guaranteed to have zero moments up to the order of the
cardinal spline functions. The semi-orthogonal (s.o.) spline-wavelets used here cannot be replaced by any other biorthogonal
wavelet (spline or otherwise) which is not s.o., since the duality principle must be applied to some subspace of the multiresolution
analysis under consideration. In contrast with the existing procedures based on direct numerical integration or an FFT-based
multi-voice per octave scheme, the computational complexity of our FIWT algorithm does not increase with the increasing number
of values of the scale parameter.
Received March 3, 1994 相似文献
20.
Holger Boche 《manuscripta mathematica》1998,95(2):137-147
The behaviour of multidimensional Shannon sampling series for continuous functions is examined. A continuous function g
1∈C
0 [0,1]2 with support in the rectangle [0,1] × [0,?] is indicated in the paper for which the two dimensional Shannon sampling series
diverge almost everywhere in the rectangle [0,1] × [?,1]. This shows that the localization principle for Shannon sampling
series cannot hold in two dimensions and in higher dimensions. The result solves a problem formulated by P.L. Butzer.
Received: 21 December 1995 / Revised version: 5 October 1996 相似文献