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171.
Summary We propose and analyse a method of estimating the poles near the unit circleT of a functionG whose values are given at a grid of points onT: we give an algorithm for performing this estimation and prove a convergence theorem. The method is to identify the phase for an estimate by considering the peaks of the absolute value ofG onT, and then to estimate the modulus by seeking a bestL
2 fit toG over a small arc by a first order rational function. These pole estimates lead to the construction of a basis ofL
2 which is well suited to the numerical representation of the Hankel operator with symbolG and thereby to the numerical solution of the Nehari problem (computing the bestH
, analytic, approximation toG relative to theL
norm), as analysed in [HY]. We present the results of numerical tests of these algorithms.Partially supported by grants from the AFOSR and NSF 相似文献
172.
Clemens Markett 《Constructive Approximation》1989,5(1):383-404
One of the most far-reaching qualities of an orthogonal system is the presence of an explicit product formula. It can be utilized to establish a convolution structure and hence is essential for the harmonic analysis of the corresponding orthogonal expansion. As yet a convolution structure for Fourier-Bessel series is unknown, maybe in view of the unpractical nature of the corresponding expanding functions called Fourier-Bessel functions. It is shown in this paper that for the half-integral values of the parameter
,n=0, 1, 2,, the Fourier-Bessel functions possess a product formula, the kernel of which splits up into two different parts. While the first part is still the well-known kernel of Sonine's product formula of Bessel functions, the second part is new and reflects the boundary constraints of the Fourier-Bessel differential equation. It is given, essentially, as a finite sum over triple products of Bessel polynomials. The representation is explicit up to coefficients which are calculated here for the first two nontrivial cases
and
. As a consequence, a positive convolution structure is established for
. The method of proof is based on solving a hyperbolic initial boundary value problem.Communicated by Tom H. Koornwinder. 相似文献
173.
Ann-sphere order is a finite partially ordered set representable by containment ofn-spheres in Euclidean (n+1)-space. We present a sequence {P
i
} of ordered sets such that eachP
i
is ann-sphere order only forni; one consequence is that we are able to determine the dimension of a Euclidean space-time manifold from the finite suborders of its causality order.Research supported by ONR grant N00014 85-K-0769. 相似文献
174.
The Huber criterion for data fitting is a combination of thel
1 and thel
2 criteria which is robust in the sense that the influence of wild data points can be reduced. We present a trust region and a Marquardt algorithm for Huber estimation in the case where the functions used in the fit are non-linear. It is demonstrated that the algorithms converge under the usual conditions. 相似文献
175.
A. Girard 《Numerische Mathematik》1989,56(1):1-23
Summary We propose a fast Monte-Carlo algorithm for calculating reliable estimates of the trace of the influence matrixA
involved in regularization of linear equations or data smoothing problems, where is the regularization or smoothing parameter. This general algorithm is simply as follows: i) generaten pseudo-random valuesw
1, ...,w
n
, from the standard normal distribution (wheren is the number of data points) and letw=(w
1, ...,w
n
)
T
, ii) compute the residual vectorw–A
w, iii) take the normalized inner-product (w
T
(w–A
w))/(w
T
w) as an approximation to (1/n)tr(I–A
). We show, both by theoretical bounds and by numerical simulations on some typical problems, that the expected relative precision of these estimates is very good whenn is large enough, and that they can be used in practice for the minimization with respect to of the well known Generalized Cross-Validation (GCV) function. This permits the use of the GCV method for choosing in any particular large-scale application, with only a similar amount of work as the standard residual method. Numerical applications of this procedure to optimal spline smoothing in one or two dimensions show its efficiency. 相似文献
176.
Much recent work has been done to investigate convergence of modified continued fractions (MCF's), following the proof by Thron and Waadeland [35] in 1980 that a limit-periodic MCFK(a
n
, 1;x
1), with
andnth approximant
相似文献
177.
We consider first the initial-boundary value problem for the parabolic equation
|