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1.
A band-dominant function on the Euclidean sphere embedded in R q+1 is the restriction to this sphere of an entire function of q+1 complex variables having a finite exponential type in each of its variables. We develop a method to represent such a function using finitely many bits, using the values of the function at scattered sites on the sphere. The number of bits required in our representation is asymptotically the same as the metric entropy of the class of such functions with respect to any of the L p norms on the sphere. 相似文献
2.
LetW
(x) be a function nonnegative inR, positive on a set of positive measure, and such that all power moments ofW
2(x) are finite. Let {p
n
(W
2;x)}
0
denote the sequence of orthonormal polynomials with respect to the weightW
2(x), and let {A
n
}
1
and {B
n
}
1
denote the coefficients in the recurrence relation
相似文献
3.
Letw be a suitable weight function,B
n,p denote the polynomial of best approximation to a functionf inL
w
p
[–1, 1],v
n
be the measure that associates a mass of 1/(n+1) with each of then+1 zeros ofB
n+1,p–B
n,p and be the arcsine measure defined by
. We estimate the rate at which the sequencev
n
converges to in the weak-* topology. In particular, our theorem applies to the zeros of monic polynomials of minimalL
w
p
norm.This author gratefully acknowledges partial support from NSA contract #A4235802 during 1992, AFSOR Grant 226113 during 1993 and The Alexander von Humboldt Foundation during both of these years. 相似文献
4.
Journal of Fourier Analysis and Applications - The problem of super-resolution in general terms is to recuperate a finitely supported measure $$\mu $$ given finitely many of its coefficients... 相似文献
5.
Approximation properties of zonal function networks using scattered data on the sphere 总被引:13,自引:0,他引:13
A zonal function (ZF) network is a function of the form x↦∑
k=1
n
c
k
(x · y
k), where x and the y
k's are on the unit sphere in q+1 dimensional Euclidean space, and where the y
k's are scattered points. In this paper, we study the degree of approximation by ZF networks. In particular, we compare this
degree of approximation with that obtained with the classical spherical harmonics. In many cases of interest, this is the
best possible for a given amount of information regarding the target function. We also discuss the construction of ZF networks
using scattered data. Our networks require no training in the traditional sense, and provide theoretically predictable rates
of approximation.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
6.
7.
R. R. Mhaskar S. E. Olson G. Raithel 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2007,41(2):221-227
A method of imaging distributions of cold atoms under the presence of large
trapping-field-induced level shifts is investigated. By utilizing a probe
laser tuned to an open transition, the fluorescence yield per atom is
largely fixed throughout the trap volume, independent of the trapping
field. This enables a reliable conversion of fluorescence images into
atomic-density profiles. The method is applied to measure distributions of
87Rb atoms in a high-gradient (2.7 kG/cm) magnetic atom guide. We
characterize the parameters for which the open-channel imaging method
performs best. Results of quantum Monte Carlo simulations verify the
underlying assumptions of the method. 相似文献
8.
We study the solutions of an equation of the form Lu=f, where L is a pseudo-differential operator defined for functions on the unit sphere embedded in a Euclidean space, f is a given function, and u is the desired solution. We give conditions under which the solution exists, and deduce local smoothness properties of u given corresponding local smoothness properties of f, measured by local Besov spaces. We study the global and local approximation properties of the spectral solutions, describe
a method to obtain approximate solutions using values of f at points on the sphere and polynomial operators, and describe the global and local rates of approximation provided by our
polynomial operators.
The research of this author was supported, in part, by grant DMS-0204704 from the National Science Foundation and grant W911NF-04-1-0339
from the U.S. Army Research Office 相似文献
9.
We, herein, report synthesis of methyl ester of (E)-5-octadecen-7,9-diynoic acid, one of the novel acetylenic fatty acids of Paramacralabium caeruleum root bark with unique biological activity, by coupling synthesized C1 to C8 fragment with 1-decyne. 相似文献
10.
Let , and for k=0,1,…, denote the orthonormalized Jacobi polynomial of degree k. We discuss the construction of a matrix H so that there exist positive constants c, c1, depending only on H, α, and β such that
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