共查询到20条相似文献,搜索用时 78 毫秒
1.
Michael I. Ganzburg 《Journal of Approximation Theory》2002,119(2):193-213
In this paper, we establish new asymptotic relations for the errors of approximation in Lp[−1,1], 0<p∞, of xλ, λ>0, by the Lagrange interpolation polynomials at the Chebyshev nodes of the first and second kind. As a corollary, we show that the Bernstein constant
is finite for λ>0 and
. 相似文献
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2.
A. Bultheel P. González-Vera E. Hendriksen O. Njåstad 《Journal of Approximation Theory》1997,89(3):344-371
In Akhiezer's book [“The Classical Moment Problem and Some Related Questions in Analysis,” Oliver & Boyd, Edinburghasol;London, 1965] the uniqueness of the solution of the Hamburger moment problem, if a solution exists, is related to a theory of nested disks in the complex plane. The purpose of the present paper is to develop a similar nested disk theory for a moment problem that arises in the study of certain orthogonal rational functions. Let {αn}∞n=0be a sequence in the open unit disk in the complex plane, let
(
/|αk|=−1 whenαk=0), and let
We consider the following “moment” problem: Given a positive-definite Hermitian inner product ·, · on
×
, find a non-decreasing functionμon [−π, π] (or a positive Borel measureμon [−π,π)) such that
In particular we give necessary and sufficient conditions for the uniqueness of the solution in the case that
If this series diverges the solution is always unique. 相似文献
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3.
Generalizations of prophet inequalities for single sequences are obtained for optimal stopping of several parallel sequences of independent random variables. For example, if {Xi, j, 1 ≤ i ≤ n, 1 ≤ j < ∞} are independent non-negative random variables, then
and this bound is best possible. Applications are made to comparisons of the optimal expected returns of various alternative methods of stopping of parallel processes. 相似文献
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4.
We consider functionals of the calculus of variations of the form F(u)= ∝01 f(x, u, u′) dx defined for u ε W1,∞(0, 1), and we show that the relaxed functional
with respect to weak W1,1(0, 1) convergence can be written as
, where the additional term L(u), called the Lavrentiev term, is explicitly identified in terms of F. 相似文献
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5.
Orthogonal expansions in product Jacobi polynomials with respect to the weight function Wα, β(x)=∏dj=1 (1−xj)αj (1+xj)βj on [−1, 1]d are studied. For αj, βj>−1 and αj+βj−1, the Cesàro (C, δ) means of the product Jacobi expansion converge in the norm of Lp(Wα, β, [−1, 1]d), 1p<∞, and C([−1, 1]d) if
Moreover, for αj, βj−1/2, the (C, δ) means define a positive linear operator if and only if δ∑di=1 (αi+βi)+3d−1. 相似文献
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6.
It is a well-known conjecture that given m ε N, the set of natural numbers, the sequence {mn}∞n−0, defined by the iterative formula m0 = m,
has some iterate mj = 1. It is shown in this paper that for any k ε N, “almost every” natural number m greater than unity has k iterates less than m. 相似文献
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7.
The best possible constant An in an inequality of Markov type
, where ·[0, ∞) denotes the sup-norm on the half real line [0, ∞) and pn is an arbitrary polynomial of degree at most n, is determined in terms of the weighted Chebyshev polynomials associated with the Laguerre weight e−x on [0, ∞). 相似文献
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8.
The nth cyclic function is defined by
We prove that if k is an integer with 1kn−1, then
holds for all positive real numbers x with the best possible constantsα=1 and β= 2n-k over n. 相似文献
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9.
A.L. Lukashov 《Journal of Approximation Theory》1998,95(3):333-352
Chebyshev–Markov rational functions are the solutions of the following extremal problem
withKbeing a compact subset of
andωn(x) being a fixed real polynomial of degree less thann, positive onK. A parametric representation of Chebyshev–Markov rational functions is found forK=[b1, b2]…[b2p−1, b2p], −∞<b1b2<…<b2p−1b2p<+∞ in terms of Schottky–Burnside automorphic functions. 相似文献
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10.
The object of this paper is to study the rapidity of convergence of the Taylor mean of the Fourier series of ƒ(x) when ƒ(x) belongs to the class Lip(α, p). We show that it is of Jackson order provided that a suitable integrability condition is imposed upon the function
. 相似文献
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11.
Jos M. Rodríguez 《Journal of Approximation Theory》2001,108(2):119
We characterize the set of functions which can be approximated by polynomials with the following norm
for a big class of weights w0, w1, …, wk 相似文献
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12.
Let p > 1, and dμ a positive finite Borel measure on the unit circle Γ: = {z ε C: ¦z¦ = 1}. Define the monic polynomial φn, p(z)=zn+…εPn >(the set of polynomials of degree at most n) satisfying
. Under certain conditions on dμ, the asymptotics of φn, p(z) for z outside, on, or inside Γ are obtained (cf. Theorems 2.2 and 2.4). Zero distributions of φn, p are also discussed (cf. Theorems 3.1 and 3.2). 相似文献
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13.
G. Min 《Journal of Approximation Theory》1999,98(2):197
This note characterizes the denseness of rational systems
in C[−1, 1], where the nonreal poles in {ak}∞k=1
\[−1, 1] are paired by complex conjugation. This extends an Achiezer's result. 相似文献
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14.
D. S. Lubinsky 《Journal of Approximation Theory》1985,44(4):343-379
Upper and lower bounds for generalized Christoffel functions, called Freud-Christoffel functions, are obtained. These have the form λn,p(W,j,x) = infPWLp(R)/|P(j)(X)| where the infimum is taken over all polynomials P(x) of degree at most n − 1. The upper and lower bounds for λn,p(W,j,x) are obtained for all 0 < p ∞ and J = 0, 1, 2, 3,… for weights W(x) = exp(−Q(x)), where, among other things, Q(x) is bounded in [− A, A], and Q″ is continuous in
β(−A, A) for some A > 0. For p = ∞, the lower bounds give a simple proof of local and global Markov-Bernstein inequalities. For p = 2, the results remove some restrictions on Q in Freud's work. The weights considered include W(x) = exp(− ¦x¦α/2), α > 0, and W(x) = exp(− exp(¦x¦)), > 0. 相似文献
15.
In this paper we consider the problem of best approximation in ℓpn, 1<p∞. If hp, 1<p<∞, denotes the best ℓp-approximation of the element h
n from a proper affine subspace K of
n, hK, then limp→∞hp=h∞*, where h∞* is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to prove that for all r
there are αj
n, 1jr, such that
, with γp(r)
n and γp(r)=
(p−r−1). 相似文献
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16.
Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables 总被引:1,自引:0,他引:1
Consider Z+d (d2)—the positive d-dimensional lattice points with partial ordering , let {Xk,kZ+d} be i.i.d. random variables with mean 0, and set Sn=∑knXk, nZ+d. We establish precise asymptotics for ∑n|n|r/p−2P(|Sn||n|1/p), and for
, (0δ1) as 0, and for
as
. 相似文献
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17.
Oblique Projections and Abstract Splines 总被引:1,自引:0,他引:1
Given a closed subspace
of a Hilbert space
and a bounded linear operator AL(
) which is positive, consider the set of all A-self-adjoint projections onto
:
In addition, if
1 is another Hilbert space, T:
→
1 is a bounded linear operator such that T*T=A and ξ
, consider the set of (T,
) spline interpolants to ξ:
A strong relationship exists between
(A,
) and sp(T,
,ξ). In fact,
(A,
) is not empty if and only if s p(T,
,ξ) is not empty for every ξ
. In this case, for any ξ
it holds
and for any ξ
, the unique vector of s p(T,
,ξ) with minimal norm is (1−PA,
)ξ, where PA,
is a distinguished element of
(A,
). These results offer a generalization to arbitrary operators of several theorems by de Boor, Atteia, Sard and others, which hold for closed range operators. 相似文献
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18.
Nancy Kopell 《Advances in Applied Mathematics》1981,2(4):389-399
We consider reaction-diffusion equations of the special type
having compact support in x. Assumptions about the relevant space scales and size of the catalytic effect exactly parallel those of Hagan (Advances in Appl. Math., 2 (1981), 400–416). The results are also parallel: For x of dimension one or two, if Ω(x) ≥ 0, Ω 0, then a unique target pattern solution which stays locally close to the homogeneous limit cycle solution. If x has dimension three, there is such a solution provided that Ω(x) is sufficiently large. Thus this paper shows that the phenomena uncovered formally by Hagan for a much larger class of kinetic equations can be rigorously substantiated for λ — ω systems. 相似文献
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19.
We consider noncommutative continued fractions of the form b0 + a1(b1 + a2(b2 + a3(…)−1 c3)−1 c2)−1 c1, (1) where an, bn and cn are elements of some Banach algebra B and bn−1 exists. Such expressions play an important role in the numerical investigation of various problems in theoretical physics and in applied mathematics, but up to now their convergence was not studied in the general case. In this paper we prove a theorem which is an extension of a wellknown theorem of Pringsheim and, in particular, guarantees the convergence of (1) under the following hypotheses:
. As an application, we give a generalization of a theorem of van Vleck. The paper closes with an extensive bibliography. 相似文献
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20.
Let 0<p<∞ and 0α<β2π. We prove that for n1 and trigonometric polynomials sn of degree n, we have
cnp ∫βα |sn(θ)|p dθ, where c is independent of α, β, n, sn. The essential feature is the uniformity in [α,β] of the estimate and the fact that as [α,β] approaches [0,2π], we recover the Lp Markov inequality. The result may be viewed as the complete Lp form of Videnskii's inequalities, improving earlier work of the second author. 相似文献
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