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《Comptes Rendus Mathematique》2008,346(23-24):1235-1238
Irregular sampling and “stable sampling” of band-limited functions have been studied by H.J. Landau [H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52]. We prove that quasicrystals are sets of stable sampling. To cite this article: B. Matei, Y. Meyer, C. R. Acad. Sci. Paris, Ser. I 346 (2008). 相似文献
3.
Qing Gu 《Proceedings of the American Mathematical Society》2000,128(10):2973-2979
The question of which groups are isomorphic to groups of interpolation maps for interpolation families of wavelet sets was raised by Dai and Larson. In this article it is shown that any finite group is isomorphic to a group of interpolation maps for some interpolation family of wavelet sets.
4.
Both barycentric Lagrange interpolation and barycentric rational interpolation are thought to be stable and effective methods for approximating a given function on some special point sets. A direct evaluation of these interpolants due to N interpolation points at M sampling points requires \(\mathcal {O}(NM)\) arithmetic operations. In this paper, we introduce two fast multipole methods to reduce the complexity to \(\mathcal {O}(\max \left \{N,M\right \})\). The convergence analysis is also presented in this paper. 相似文献
5.
This paper is devoted to studying the problem of optimal recovery for Sobolev function classesW
2
r
(ℝ) inL
2 (ℝ) by using the information of function values on denumerable points. Forr≥1, we determine that the optimal sampling points are arranged equidistantly in a suitable collection of sets of sampling
points and find two kinds of cardinal spline interpolation as optimal algorithms.
This author is supported by the National Natural Science Foundation of China. 相似文献
6.
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
7.
W. Norrie Everitt Antonio G. García Miguel Angel Hernández-Medina 《Results in Mathematics》2008,51(3-4):215-228
The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. A challenging problem
is to characterize the situations when these sampling formulas can be written as Lagrange-type interpolation series. This
article gives a necessary and sufficient condition to ensure that when the sampling formula is associated with an analytic
Kramer kernel, then it can be expressed as a quasi Lagrange-type interpolation series; this latter form is a minor but significant
modification of a Lagrange-type interpolation series. Finally, a link with the theory of de Branges spaces is established.
Received: October 8, 2007. Revised: December 13, 2007. 相似文献
8.
Xavier Dahan 《Journal of Complexity》2012,28(1):109-135
We give bit-size estimates for the coefficients appearing in triangular sets describing positive-dimensional algebraic sets defined over Q. These estimates are worst case upper bounds; they depend only on the degree and height of the underlying algebraic sets. We illustrate the use of these results in the context of a modular algorithm.This extends the results by the first and the last author, which were confined to the case of dimension 0. Our strategy is to get back to dimension 0 by evaluation and interpolation techniques. Even though the main tool (height theory) remains the same, new difficulties arise to control the growth of the coefficients during the interpolation process. 相似文献
9.
A class of sets correct for multivariate polynomial interpolation is defined and verified, and shown to coincide with the collection of all correct sets constructible by the recursive application of Radon’s recipe, and a recent concrete recipe for correct sets is shown to produce elements in that class. 相似文献
10.
Stefano De Marchi Robert Schaback Holger Wendland 《Advances in Computational Mathematics》2005,23(3):317-330
The goal of this paper is to construct data-independent optimal point sets for interpolation by radial basis functions. The interpolation points are chosen to be uniformly good for all functions from the associated native Hilbert space. To this end we collect various results on the power function, which we use to show that good interpolation points are always uniformly distributed in a certain sense. We also prove convergence of two different greedy algorithms for the construction of near-optimal sets which lead to stable interpolation. Finally, we provide several examples.
AMS subject classification 41A05, 41063, 41065, 65D05, 65D15This work has been done with the support of the Vigoni CRUI-DAAD programme, for the years 2001/2002, between the Universities of Verona and Göttingen. 相似文献
11.
This is the third part of a note on multivariate interpolation. Some remainder formulas for interpolation on knot sets that are perspective images of standard lower data sets are given. They apply to all knot systems considered in parts I and II.Partially supported by PS900121. 相似文献
12.
Chen Dirong 《数学年刊B辑(英文版)》1992,13(4):396-405
The author obtains the excat values of the average n-K widths for some Sobolev classes defined by an ordinary differential operator $\[P(D) = \prod\limits_{i = 1}^r {D - {t_i}I),{t_i} \in R} \]$,in the metric L_p(R),$1\leq p\leq \infty$,and identifies some optimal subspaces.Furthermore,the optimal interpolation problem for these Sobolev classes is considered by sampling the function values at some countable sets of points distributed reasonably on R,and some exact results are obtained. 相似文献
13.
《Journal of Computational and Applied Mathematics》2012,236(6):1656-1666
Multivariate Birkhoff interpolation is the most complicated polynomial interpolation problem and the theory about it is far from systematic and complete. In this paper we derive an Algorithm B-MB (Birkhoff-Monomial Basis) and prove B-MB giving the minimal interpolation monomial basis w.r.t. the lexicographical order of the multivariate Birkhoff problem. This algorithm is the generalization of Algorithm MB in [L. Cerlinco, M. Mureddu, From algebraic sets to monomial linear bases by means of combinatorial algorithms, Discrete Math. 139 (1995) 73–87] which is a well known fast algorithm used to compute the interpolation monomial basis of the Hermite interpolation problem. 相似文献
14.
Na LeiJunjie Chai Peng XiaYing Li 《Journal of Computational and Applied Mathematics》2011,236(6):1656-1666
Multivariate Birkhoff interpolation is the most complicated polynomial interpolation problem and the theory about it is far from systematic and complete. In this paper we derive an Algorithm B-MB (Birkhoff-Monomial Basis) and prove B-MB giving the minimal interpolation monomial basis w.r.t. the lexicographical order of the multivariate Birkhoff problem. This algorithm is the generalization of Algorithm MB in [L. Cerlinco, M. Mureddu, From algebraic sets to monomial linear bases by means of combinatorial algorithms, Discrete Math. 139 (1995) 73-87] which is a well known fast algorithm used to compute the interpolation monomial basis of the Hermite interpolation problem. 相似文献
15.
A method is presented to construct interpolation functions intothe 2 x 2 open spectral unit ball. For the spectral NevanlinnaPickproblem, these functions are in some sense extremal, and theset of all these interpolation functions is enough to solveany interpolation problem, with solvable finite interpolationdata. This fact is used to compute the complex geodesics forthe symmetrized bidisc and for the spectral unit ball, and tosolve completely the two-point interpolation problem for thetwo target sets. 相似文献
16.
Jorge Galindo 《Advances in Mathematics》2004,188(1):51-68
Rosenthal's theorem describing those Banach spaces containing no copy of ?1 is extended to topological groups replacing ?1-basis by interpolation sets in the sense of Hartman and Ryll-Nardzewsky (Colloq. Math. 12 (1964) 23-39). This extension provides a characterization of those locally compact groups containing no interpolation sets and of those locally compact groups which respect compactness, i.e, such that every Bohr compact subset is compact. The approach followed in this paper sheds some light on other questions related to the duality theory of non-Abelian locally compact groups. 相似文献
17.
On Rational Interpolation to |x| 总被引:1,自引:0,他引:1
We consider Newman-type rational interpolation to |x| induced by arbitrary sets of interpolation nodes, and we show that under mild restrictions on the location of the interpolation
nodes, the corresponding sequence of rational interpolants converges to |x|.
Date received: August 18, 1995. Date revised: January 10, 1996. 相似文献
18.
A. M. Kotochigov 《Journal of Mathematical Sciences》2005,129(4):4022-4039
We consider the problem of free interpolation for the spaces of analytic functions with derivative of order s in the Hardy space Hp. For the sets that satisfy the Stolz condition, we obtain a condition necessary for interpolation: if 1 ≤ p < ∞, then the set must be a union of s sparse sets. For p = ∞, we obtain a necessary and sufficient condition for interpolation: the set must be a union of s + 1 sparse sets. In this case, we construct an extension operator. Bibliography 11 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 169–202. 相似文献
19.
L. P. Bolgiano Jr. 《Numerical Functional Analysis & Optimization》2013,34(4):323-329
The infinite Gregory-Newton series is shown capable of interpolating unbandlimited signals which have the form of the natural responses of linear systems. Like the Whittaker cardinal function interpolation, this interpolation requires the sampling rate to satisfy a certain criterion. This criterion differs from the Nyquist criterion in that the signals are characterized by pole frequencies, instead of maximum frequencies, which must be less than a submultiple of the sampling rate. 相似文献
20.
O.V. Davydov G. Nürnberger F. Zeilfelder 《Journal of Computational and Applied Mathematics》1998,90(2):209-134
By using the algorithm of Nürnberger and Riessinger (1995), we construct Hermite interpolation sets for spaces of bivariate splines Sqr(Δ1) of arbitrary smoothness defined on the uniform type triangulations. It is shown that our Hermite interpolation method yields optimal approximation order for q 3.5r + 1. In order to prove this, we use the concept of weak interpolation and arguments of Birkhoff interpolation. 相似文献