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1.
D. S. Lubinsky 《Constructive Approximation》1991,7(1):501-519
We show that the most of the time, most poles of diagonal multipoint Padé or best rational approximants to functions admitting fast rational approximation, leave the region of meromorphy. Following is a typical result: Letf be single-valued and analytic in CS, where cap(S)=0. Let {n
j
}
j=1
be an increasing sequence of positive integers withn
j+1/n
j
1 asj. Then there exists an infinite sequenceL of positive integers such that asj,jL the total multiplicity of poles of any sequence of type (n
j
,n
j
) multipoint Padé or best rational approximants tof, iso(n
j
) in any compactK in whichf is meromorphic. The sequenceL is independent of the particular sequence of multipoint Padé or best approximants, and yields the same behavior for near-best approximants. If the errors of best approximation on some compact set satisfy a weak regularity condition, then we may takeL={1,2,3,}.Communicated by Edward B. Saff. 相似文献
2.
We consider rational approximations of the form
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3.
It is well known that best complex rational Chebyshev approximants are not always unique and that, in general, they cannot be characterized by the necessary local Kolmogorov condition or by the sufficient global Kolmogorov condition. Recently, Ruttan (1985) proposed an interesting sufficient optimality criterion in terms of positive semidefiniteness of some Hermitian matrix. Moreover, he asserted that this condition is also necessary, and thus provides a characterization of best approximants, in a fundamental case.In this paper we complement Ruttan's sufficient optimality criterion by a uniqueness condition and we present a simple procedure for computing the set of best approximants in case of nonuniqueness. Then, by exhibiting an approximation problem on the unit disk, we point out that Ruttan's characterization in the fundamental case is not generally true. Finally, we produce several examples of best approximants on a real interval and on the unit circle which, among other things, give some answers to open questions raised in the literature. 相似文献
4.
In this paper, characterizations for lim
n(R
n
(f)/(n
–1)=0 inH
and for lim
n(n
r+
R
n
(f)=0 inW
r Lip ,r1, are given, while, forZ, a generalization to a related result of Newman is established.Communicated by Ronald A. DeVore. 相似文献
5.
The main result proved in the paper is: iff is absolutely continuous in (–, ) andf' is in the real Hardy space ReH
1, then
for everyn1, whereR
n(f) is the best uniform approximation off by rational functions of degreen. This estimate together with the corresponding inverse estimate of V. Russak [15] provides a characterization of uniform rational approximation.Communicated by Ronald A. DeVore. 相似文献
6.
The paper deals with rational approximation over the real Hardy spaceH
2, R(V), whereV is the complement of the closed unit disk. The results concern Stieltjes functions
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