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1.
We show that ifw(x)=exp(–|x|), then in the case =1 for every continuousf that vanishes outside the support of the corresponding extremal measure there are polynomialsP
n of degree at mostn such thatw
n
P
n uniformly tends tof, and this is not true when <1. these=" are=" the=" missing=" cases=" concerning=" approximation=" by=" weighted=" polynomials=" of=" the=">1.>w
n
P
n wherew is a Freud weight. Our second theorem shows that even if we are only interested in approximation off on the extremal support, the functionf must still vanish at the endpoints, and we actually determine the (sequence of) largest possible intervals where approximation is possible. We also briefly discuss approximation by weighted polynomials of the formW(anx)P
n
(x).Communicated by Edward B. Saff. 相似文献
2.
In this article, an iterative method for the approximate solution to one‐dimensional variable‐coefficient Burgers' equation is proposed in the reproducing kernel space W(3,2). It is proved that the approximation wn(x,t) converges to the exact solution u(x,t) for any initial function w0(x,t) ε W(3,2), and the approximate solution is the best approximation under a complete normal orthogonal system . Moreover the derivatives of wn(x,t) are also uniformly convergent to the derivatives of u(x,t).© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 相似文献
3.
In 1998, Kleinbock and Margulis proved Sprindzuk’s conjecture pertaining to metrical Diophantine approximation (and indeed
the stronger Baker–Sprindzuk conjecture). In essence, the conjecture stated that the simultaneous homogeneous Diophantine
exponent w
0(x) = 1/n for almost every point x on a nondegenerate submanifold M \mathcal{M} of
\mathbbRn {\mathbb{R}^n} . In this paper, the simultaneous inhomogeneous analogue of Sprindzuk’s conjecture is established. More precisely, for any
“inhomogeneous” vector θ ∈
\mathbbRn {\mathbb{R}^n} we prove that the simultaneous inhomogeneous Diophantine exponent w
0(x
,
θ) is 1/n for almost every point x on M \mathcal{M} . The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent w
0(x) is 1/n for almost all x ∈ M \mathcal{M} if and only if, for any θ ∈
\mathbbRn {\mathbb{R}^n} , the inhomogeneous exponent w
0(x
,
θ) = 1/n for almost all x ∈ M \mathcal{M} . The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered
by us. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront
the main ideas while omitting the abstract and technical notions that come with describing the inhomogeneous transference
principle in all its glory. 相似文献
4.
Thomas Kühn Hans-Gerd Leopold Winfried Sickel Leszek Skrzypczak 《Mathematische Zeitschrift》2007,255(1):1-15
We determine the exact asymptotic order of the entropy numbers of compact embeddings of weighted Besov spaces in the case where the ratio of the weights w(x) = w
1(x)/w
2(x) is of logarithmic type. This complements the known results for weights of polynomial type. The estimates are given in terms of the number 1/p = 1/p
1 − 1/p
2 and the function w(x). We find an interesting new effect: if the growth rate at infinity of w(x) is below a certain critical bound, then the entropy numbers depend only on w(x) and no longer on the parameters of the two Besov spaces. All results remain valid for Triebel–Lizorkin spaces as well. 相似文献
5.
Edward A. Bender E. Rodney Canfield Brendan D. McKay 《Random Structures and Algorithms》1990,1(2):127-169
Let c(n, q) be the number of connected labeled graphs with n vertices and q ≤ N = (2n ) edges. Let x = q/n and k = q ? n. We determine functions wk ? 1. a(x) and φ(x) such that c(n, q) ? wk(qN)enφ(x)+a(x) uniformly for all n and q ≥ n. If ? > 0 is fixed, n→ ∞ and 4q > (1 + ?)n log n, this formula simplifies to c(n, q) ? (Nq) exp(–ne?2q/n). on the other hand, if k = o(n1/2), this formula simplifies to c(n, n + k) ? 1/2 wk (3/π)1/2 (e/12k)k/2nn?(3k?1)/2. 相似文献
6.
H.A. Aimar A.L. Bernardis F.J. Martín-Reyes 《Journal of Fourier Analysis and Applications》2003,9(5):497-510
We study boundedness and convergence on L
p
(R
n
,d) of the
projection operators P
j
given by MRA structures with non-necessarily
compactly supported scaling function. As a consequence, we prove that if
w is a locally integrable function such that w
-(1/p–1)(x)
(1+|x|)-N
is integrable for some N > 0, then the Muckenhoupt A
p
condition is necessary and sufficient for the associated wavelet system to
be an unconditional basis for the weighted space L
p
(R
n
,w(x) dx),
1 < p < . 相似文献
7.
We study the family of divergence-type second-order parabolic equations
we(x)\frac?u?t=div(a(x)we(x) ?u), x ? \mathbbRn{\omega_\varepsilon(x)\frac{\partial u}{\partial t}={\rm div}(a(x)\omega_\varepsilon(x) \nabla u), x \in \mathbb{R}^n} , with parameter ${\varepsilon >0 }${\varepsilon >0 } , where a(x) is uniformly elliptic matrix and we=1{\omega_\varepsilon=1} for x
n
< 0 and we=e{\omega_\varepsilon=\varepsilon} for x
n
> 0. We show that the fundamental solution obeys the Gaussian upper bound uniformly with respect to e{\varepsilon} . 相似文献
8.
Zheng Zukang 《高校应用数学学报(英文版)》2004,19(1):90-100
An algorithm of continuous stage-space MCMC method for solving algebra equation f(x)=0 is given. It is available for the case that the sign of f(x) changes frequently or the derivative f′(x) does not exist in the neighborhood of the root, while the Newton method is hard to work. Let n be the number of random variables created by computer in our algorithm. Then after m=O(n) transactions from the initial value x
0,x* can be got such that |f(x*)|<e
−cm |f(x
0)| by choosing suitable positive constant c. An illustration is also given with the discussion of convergence by adjusting the parameters in the algorithm.
Supported by the National Natural Science Foundation of China (70171008). 相似文献
9.
S. P. Zhou 《Israel Journal of Mathematics》1992,78(1):75-83
The present paper gives a converse result by showing that there exists a functionf ∈C
[−1,1], which satisfies that sgn(x)f(x) ≥ 0 forx ∈ [−1, 1], such that {fx75-1} whereE
n
(0)
(f, 1) is the best approximation of degreen tof by polynomials which are copositive with it, that is, polynomialsP withP(x(f(x) ≥ 0 for allx ∈ [−1, 1],E
n(f) is the ordinary best polynomial approximation off of degreen. 相似文献
10.
We investigate two problems concerning uniform approximation by weighted rationals {w nrn~ ∞ n=1 }, wherer n=pn Namely, forw(x):=e x we prove that uniform convergence to 1 ofw nrn is not possible on any interval [0,a] witha>2π. Forw(x):=x ?, ?>1, we show that uniform convergence to 1 ofw nrn is not possible on any interval [b, 1] withb<tan 4(π(??1)/4?). (The latter result can be interpreted as a rational analogue of results concerning “incomplete polynomials.”) More generally, for α≥0, β≥0, α+β>0, we investigate forw(x)=e x andw(x)=x ?, the possibility of approximation byw n pn/qn~ ∞ n=1 , where depp n≤αn, degq n≤βn. The analysis utilizes potential theoretic methods. These are essentially sharp results though this will not be established in this paper. 相似文献
11.
Let 𝒯(n,?r;?W n?1) be the set of all n-vertex weighted trees with r vertices of degree 2 and fixed positive weight set W n?1, 𝒫(n,?γ;?W n?1) the set of all n-vertex weighted trees with q pendants and fixed positive weight set W n?1, where W n?1?=?{w 1,?w 2,?…?,?w n?1} with w 1???w 2???···???w n?1?>?0. In this article, we first identify the unique weighted tree in 𝒯(n,?r;?W n?1) with the largest adjacency spectral radius. Then we characterize the unique weighted trees with the largest adjacency spectral radius in 𝒫(n,?γ;?W n?1). 相似文献
12.
Let I be a finite or infinite interval and dμ a measure on I. Assume that the weight function w(x)>0, w′(x) exists, and the function w′(x)/w(x) is non-increasing on I. Denote by ℓk's the fundamental polynomials of Lagrange interpolation on a set of nodes x1<x2<<xn in I. The weighted Lebesgue function type sum for 1≤i<j≤n and s≥1 is defined byIn this paper the exact lower bounds of Sn(x) on a “big set” of I and are obtained. Some applications are also given. 相似文献
13.
Kenji Kimura 《Discrete Applied Mathematics》2010,158(18):2071-2074
We extend the notion of a defensive alliance to weighted graphs. Let (G,w) be a weighted graph, where G is a graph and w be a function from V(G) to the set of positive real numbers. A non-empty set of vertices S in G is said to be a weighted defensive alliance if ∑x∈NG(v)∩Sw(x)+w(v)≥∑x∈NG(v)−Sw(x) holds for every vertex v in S. Fricke et al. (2003) [3] have proved that every graph of order n has a defensive alliance of order at most . In this note, we generalize this result to weighted defensive alliances. Let G be a graph of order n. Then we prove that for any weight function w on V(G), (G,w) has a defensive weighted alliance of order at most . We also extend the notion of strong defensive alliance to weighted graphs and generalize a result in Fricke et al. (2003) [3]. 相似文献
14.
Dong Hyun Cho 《Czechoslovak Mathematical Journal》2009,59(2):431-452
Let C[0, T] denote the space of real-valued continuous functions on the interval [0, T] with an analogue w
ϕ of Wiener measure and for a partition 0 = t
0 < t
1 < ... < t
n
< t
n+1 = T of [0, T], let X
n
: C[0, T] → ℝ
n+1 and X
n+1: C[0, T] → ℝ
n+2 be given by X
n
(x) = (x(t
0), x(t
1), ..., x(t
n
)) and X
n+1(x) = (x(t
0), x(t
1), ..., x(t
n+1)), respectively.
In this paper, using a simple formula for the conditional w
ϕ-integral of functions on C[0, T] with the conditioning function X
n+1, we derive a simple formula for the conditional w
ϕ-integral of the functions with the conditioning function X
n
. As applications of the formula with the function X
n
, we evaluate the conditional w
ϕ-integral of the functions of the form F
m
(x) = ∫0
T
(x(t))
m
for x ∈ C[0, T] and for any positive integer m. Moreover, with the conditioning X
n
, we evaluate the conditional w
ϕ-integral of the functions in a Banach algebra
which is an analogue of the Cameron and Storvick’s Banach algebra
. Finally, we derive the conditional analytic Feynman w
ϕ-integrals of the functions in
.
相似文献
15.
We show that the Poisson maximal operator for the tube over the light-cone, P
*, is bounded in the weighted space L
p
(w) if and only if the weight w(x) belongs to the Muckenhoupt class A
p
. We also characterize with a geometric condition related to the intrinsic geometry of the cone the weights v(x) for which P
* is bounded from L
p
(v) into L
p
(u), for some other weight u(x) > 0. Some applications to a.e. restricted convergence of Poisson integrals are given. 相似文献
16.
S. N. M. Ruijsenaars 《Indagationes Mathematicae》2003,14(3-4):515
In earlier work we introduced and studied two commuting generalized Lamé operators, obtaining in particular joint eigenfunctions for a dense set in the natural parameter space. Here we consider these difference operators and their eigenfunctions in relation to the Hilbert space L2((0, π/r), w(x)dx), with r > 0 and the weight function w(x) a ratio of elliptic gamma functions. In particular, we show that the previously known pairwise orthogonal joint eigenfunctions need only be supplemented by finitely many new ones to obtain an orthogonal base. This completeness property is derived by exploiting recent results on the large-degree Hilbert space asymptotics of a class of orthonormal polynomials. The polynomials pn(cos(rx)), n ε
, that are relevant in the Lamé setting are orthonormal in L2((0, π/r), wP(x)dx), with wp(x) closely related to w(x). 相似文献
17.
We present a class of functions gK(w), K ≥ 2, for which the recursive sequences wn + 1 = gK(wn) converge to N1/v with relative error . Newton's method results when K = 2. The coefficients of the gK(w) form a triangle, which is Pascal's for v = 2. In this case, if w1 = x1/y1, where x1, y1 is the first positive solution of Pell's equation x2 ? Ny2 = 1, then wn + 1 = xn + 1/yn + 1 is the Knpth or 2Knpth convergent of the continued fraction for , its period p being even or odd. 相似文献
18.
Vilmos Totik 《Constructive Approximation》2000,16(2):261-281
It is proven that if Q is convex and w(x)= exp(-Q(x)) is the corresponding weight, then every continuous function that vanishes outside the support of the extremal measure associated
with w can be uniformly approximated by weighted polynomials of the form w
n
P
n
. This solves a problem of P. Borwein and E. B. Saff. Actually, a similar result is true locally for any parts of the extremal
support where Q is convex.
February 10, 1998. Date revised: July 23, 1998. Date accepted: August 17, 1998. 相似文献
19.
Using a multidimensional analog of the logarithmic residue, equations are derived expressing the coefficients of the power series of implicit functionsx
j
=j(w)=j(w1,...,wm), j=1,...,n, defined by the system of equations fj(w, x)=Fj (w1,...,wm:z1,...,x
n
)=0, j=1,...,n,f
j
, (0, 0)=0, Fj(0, 0)/zk=jk in a neighborhood of the point (0, 0)C
(w,x)
m+n
, in terms of the coefficients of the power series of the functions Fj(w, z), j=1, ..., n. As a corollary, well-known formulas are obtained for the inversion of multiple power series.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 47–54, January, 1978. 相似文献
20.
A theory T is called almost ??0-categorical if for any pure types p1(x1),…,pn(xn) there are only finitely many pure types which extend p1(x1) ∪…∪pn(xn). It is shown that if T is an almost ??0-categorical theory with I(??0,T) = 3, then a dense linear ordering is interpretable in T. 相似文献