共查询到20条相似文献,搜索用时 828 毫秒
1.
R 《Journal of Approximation Theory》1993,74(3)
In this paper, we determine the exact value of average n − K width
n(Wrpq(R), Lq(R)) of Sobolev-Wiener class Wrpq(R) in the metric Lq(R) for 1 > q ≥ p > ∞ and get the value of
n(Wrp(R), Lqp(R)) for the dual case. We also solve the optimal interpolation problems of Wrpq(R) in the metric Lq(R) and Wrp(R) in the metric Lqp(R) for 1 < q ≤ p < ∞. 相似文献
2.
B. Ricceri 《Mathematical and Computer Modelling》2000,32(11-13)
In this paper, we consider a problem of the type −Δu = λ(f(u) + μg(u)) in Ω, u¦∂Ω = 0, where Ω Rn is an open-bounded set, f, g are continuous real functions on R, and λ, μ ε R. As an application of a new approach to nonlinear eigenvalues problems, we prove that, under suitable hypotheses, if ¦μ¦ is small enough, then there is some λ > 0 such that the above problem has at least three distinct weak solutions in W01,2(Ω). 相似文献
3.
Let
be the classical middle-third Cantor set and let μ
be the Cantor measure. Set s = log 2/log 3. We will determine by an explicit formula for every point x
the upper and lower s-densities Θ*s(μ
, x), Θ*s(μ
, x) of the Cantor measure at the point x, in terms of the 3-adic expansion of x. We show that there exists a countable set F
such that 9(Θ*s(μ
, x))− 1/s + (Θ*s(μ
, x))− 1/s = 16 holds for x
\F. Furthermore, for μC almost all x, Θ*s(μ
, X) − 2 · 4− s and Θ*s(μ
, x) = 4− s. As an application, we will show that the s-dimensional packing measure of the middle-third Cantor set
is 4s. 相似文献
4.
Summability of spherical h-harmonic expansions with respect to the weight function ∏j=1d |xj|2κj (κj0) on the unit sphere Sd−1 is studied. The main result characterizes the critical index of summability of the Cesàro (C,δ) means of the h-harmonic expansion; it is proved that the (C,δ) means of any continuous function converge uniformly in the norm of C(Sd−1) if and only if δ>(d−2)/2+∑j=1d κj−min1jd κj. Moreover, it is shown that for each point not on the great circles defined by the intersection of the coordinate planes and Sd−1, the (C,δ) means of the h-harmonic expansion of a continuous function f converges pointwisely to f if δ>(d−2)/2. Similar results are established for the orthogonal expansions with respect to the weight functions ∏j=1d |xj|2κj(1−|x|2)μ−1/2 on the unit ball Bd and ∏j=1d xjκj−1/2(1−|x|1)μ−1/2 on the simplex Td. As a related result, the Cesàro summability of the generalized Gegenbauer expansions associated to the weight function |t|2μ(1−t2)λ−1/2 on [−1,1] is studied, which is of interest in itself. 相似文献
5.
If T is any bounded linear operator on Besov spaces Bpσj,qj(Rn)(j=0,1, and 0<σ1<σ<σ0), it is proved that the commutator [T,Tμ]=TTμ−TμT is bounded on Bpσ,q(Rn), if Tμ is a Fourier multiplier such that μ is any (possibly unbounded) symbol with uniformly bounded variation on dyadic coronas. 相似文献
6.
Let X1,…, Xn be i.i.d. random variables symmetric about zero. Let Ri(t) be the rank of |Xi − tn−1/2| among |X1 − tn−1/2|,…, |Xn − tn−1/2| and Tn(t) = Σi = 1nφ((n + 1)−1Ri(t))sign(Xi − tn−1/2). We show that there exists a sequence of random variables Vn such that sup0 ≤ t ≤ 1 |Tn(t) − Tn(0) − tVn| → 0 in probability, as n → ∞. Vn is asymptotically normal. 相似文献
7.
This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)1 be a measurable function defined on a domain ΩRn, n2, and such that exp(βK(x))Lloc1(Ω), β>0. We show that there exist two universal constants c1(n),c2(n) with the following property: Let f be a mapping in Wloc1,1(Ω,Rn) with |Df(x)|nK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L1 log−c1(n)βL. Then automatically J(x,f) is locally in L1 logc2(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings. 相似文献
8.
Krzysztof Przes
awski 《Journal of Approximation Theory》1996,85(3):288-296
It is shown that for each convex bodyARnthere exists a naturally defined family
AC(Sn−1) such that for everyg
A, and every convex functionf: R→Rthe mappingy∫Sn−1 f(g(x)−y, x) dσ(x) has a minimizer which belongs toA. As an application, approximation of convex bodies by balls with respect toLpmetrics is discussed. 相似文献
9.
In this paper, we determine the X-inner automorphisms of the smash product R # U(L) of a prime ring R by the universal enveloping algebra U(L) of a characteristic 0 Lie algebra L. Specifically, we show that any such automorphism σ stabilizing R can be written as a product σ = σ1σ2, where σ1 is induced by conjugation by a unit of Q3(R), the symmetric Martindale ring of quotients of R, and σ2 is induced by conjugation by a unit of Q3(T). Here S = Ql(R) is the left Martindale ring of quotients of R and T is the centralizer of S in S # U(L) - R # U(L). One of the subtleties of the proof is that we must work in several unrelated overrings of R # U(L). 相似文献
10.
Yasunori Fujikoshi 《Journal of multivariate analysis》2000,72(2):249
Suppose that a nonnegative statistic T is asymptotically distributed as a chi-squared distribution with f degrees of freedom, χ2f, as a positive number n tends to infinity. Bartlett correction T was originally proposed so that its mean is coincident with the one of χ2f up to the order O(n−1). For log-likelihood ratio statistics, many authors have shown that the Bartlett corrections are asymptotically distributed as χ2f up to O(n−1), or with errors of terms of O(n−2). Bartlett-type corrections are an extension of Bartlett corrections to other statistics than log-likelihood ratio statistics. These corrections have been constructed by using their asymptotic expansions up to O(n−1). The purpose of the present paper is to propose some monotone transformations so that the first two moments of transformed statistics are coincident with the ones of χ2f up to O(n−1). It may be noted that the proposed transformations can be applied to a wide class of statistics whether their asymptotic expansions are available or not. A numerical study of some test statistics that are not a log-likelihood ratio statistic is discribed. It is shown that the proposed transformations of these statistics give a larger improvement to the chi-squared approximation than do the Bartlett corrections. Further, it is seen that the proposed approximations are comparable with the approximation based on an Edgeworth expansion. 相似文献
11.
In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is yt = Bzt + vt. The unobserved error sequence {vt} is a sequence of martingale differences with conditional covariance matrices {Σt} and satisfying supt=1,…, n
{v′tvtI(v′tvt>a) |zt, vt−1, zt−1, …}
0 as a → ∞. The sample covariance of the independent variables z1, …, zn, is assumed to have a probability limit M, constant and nonsingular; maxt=1,…,nz′tzt/n
0. If (1/n)Σt=1nΣt
Σ, constant, then √nvec(
n−B)
N(0,M−1Σ) and
n
Σ. The autoregression model is xt = Bxt − 1 + vt with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {vt}, and (1/n)Σt=max(r,s)+1(Σtvt−1−rv′t−1−s)
δrs(ΣΣ), where δrs is the Kronecker delta. Then √nvec(
n−B)
N(0,Γ−1Σ), where Γ = Σs = 0∞BsΣ(B′)s. 相似文献
12.
Consider a Hilbert space
equipped with a time-structure, i.e., a resolution E of the identity on
defined on subsets of some linearly ordered set Λ. For which x and y in
is it possible to find a causal (time respecting) compact operator T, so that Tx = y? When T is required to be a Hilbert-Schmidt operator and (Λ, E) is sufficiently regular, this question is answered in terms of the “time-densities” of x and y. The condition is that the integral ∝gLμx({s t})−1 dμy(t) should be finite, where μx and μy are the measures on Λ given by μx(Ω) = ¦|E(Ω)x¦|2 and μy(Ω) = ¦|E(Ω)y¦|2. Further a solution is given for the related problem of minimizing the sum of ¦|Tx − y¦|2 and the squared Hilbert-Schmidt norm ¦|R¦|22 of T. 相似文献
13.
Cheng He 《Journal of Mathematical Analysis and Applications》1997,210(2):512
We construct a class of weak solutions to the Navier–Stokes equations, which have second order spatial derivatives and one order time derivatives, ofppower summability for 1 < p ≤ 5/4. Meanwhile, we show thatu Ls(0, T; W2, r(Ω)) with 1/s + 3/2r = 2 for 1 < r ≤ 5/4.rcan be relaxed not to exceed 3/2 if we consider only in the interior of Ω. In the end, we extend the classical regularity theorem. Our results show thatuis a regular solution if u Ls(0, T; Lr(Ω)) with 1/s + 3/2r = 1 for Ω satisfying (1.3), with 1/s + 1/r = 5/6 for arbitrary domain inR3and 1 < s ≤ 2. For Ω = Rnwithn ≥ 3, this result was previously obtained byH. Beirão da Veiga (Chinese Ann. Math. Ser. B16, 1995, 407–412). 相似文献
14.
In a sequence ofn independent random variables the pdf changes fromf(x, 0) tof(x, 0 + δvn−1) after the firstnλ variables. The problem is to estimateλ (0, 1 ), where 0 and δ are unknownd-dim parameters andvn → ∞ slower thann1/2. Letn denote the maximum likelihood estimator (mle) ofλ. Analyzing the local behavior of the likelihood function near the true parameter values it is shown under regularity conditions that ifnn2(− λ) is bounded in probability asn → ∞, then it converges in law to the timeT(δjδ)1/2 at which a two-sided Brownian motion (B.M.) with drift1/2(δ′Jδ)1/2ton(−∞, ∞) attains its a.s. unique minimum, whereJ denotes the Fisher-information matrix. This generalizes the result for small change in mean of univariate normal random variables obtained by Bhattacharya and Brockwell (1976,Z. Warsch. Verw. Gebiete37, 51–75) who also derived the distribution ofTμ forμ > 0. For the general case an alternative estimator is constructed by a three-step procedure which is shown to have the above asymptotic distribution. In the important case of multiparameter exponential families, the construction of this estimator is considerably simplified. 相似文献
15.
We study the complexity of Fredholm problems (I−Tk)u=f of the second kind on Id=[0,1]d, where Tk is an integral operator with kernel k. Previous work on the complexity of this problem has assumed either that we had complete information about k or that k and f had the same smoothness. In addition, most of this work has assumed that the information about k and f was exact. In this paper, we assume that k and f have different smoothness; more precisely, we assume that fWr,p(Id) with r>d/p and that kWs,∞(I2d) with s>0. In addition, we assume that our information about k and f is contaminated by noise. We find that the nth minimal error is Θ(n−μ+δ), where μ=min{r/d,s/(2d)} and δ is a bound on the noise. We prove that a noisy modified finite element method has nearly minimal error. This algorithm can be efficiently implemented using multigrid techniques. We thus find tight bounds on the -complexity for this problem. These bounds depend on the cost c(δ) of calculating a δ-noisy information value. As an example, if the cost of a δ-noisy evaluation is proportional to δ−t, then the -complexity is roughly (1/)t+1/μ. 相似文献
16.
Dalibor Volný Benjamin Weiss 《Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques》2004,40(6):771-778
Let T be an ergodic automorphism of a probability space, f a bounded measurable function, . It is shown that the property that the probabilities μ(|Sn(f)|>n) are of order n−p roughly corresponds to the existence of an approximation in L∞ of f by functions (coboundaries) g−g○T, gLp. Similarly, the probabilities μ(|Sn(f)|>n) are exponentially small iff f can be approximated by coboundaries g−g○T where g have finite exponential moments.
Résumé
Soit T un automorphisme ergodique d'un espace probabilisé, f une fonction bornée mesurable et . Une correspondance est établie entre l'existence de l'estimation des probabilités μ(|Sn(f)|>n) d'ordre n−p et l'existence de l'approximation dans L∞ de la fonction f par des cobords g−g○T où g est “presque” dans Lp. De manière similaire, les probabilités μ(|Sn(f)|>n) sont d'ordre e−cn, pour un certain c>0, n=1,2… , si et seulement si f admet une approximation dans L∞ par des cobords g−g○T avec g ayant des moments exponentiels. 相似文献17.
Let T = {T(t)}t ≥ 0 be a C0-semigroup on a Banach space X. In this paper, we study the relations between the abscissa ωLp(T) of weak p-integrability of T (1 ≤ p < ∞), the abscissa ωpR(A) of p-boundedness of the resolvent of the generator A of T (1 ≤ p ≤ ∞), and the growth bounds ωβ(T), β ≥ 0, of T. Our main results are as follows.
- 1. (i) Let T be a C0-semigroup on a B-convex Banach space such that the resolvent of its generator is uniformly bounded in the right half plane. Then ω1 − ε(T) < 0 for some ε > 0.
- 2. (ii) Let T be a C0-semigroup on Lp such that the resolvent of the generator is uniformly bounded in the right half plane. Then ωβ(T) < 0 for all β>¦1/p − 1/p′¦, 1/p + 1/p′ = 1.
- 3. (iii) Let 1 ≤ p ≤ 2 and let T be a weakly Lp-stable C0-semigroup on a Banach space X. Then for all β>1/p we have ωβ(T) ≤ 0.
18.
Letμbe a Gaussian measure (say, onRn) and letK,LRnbe such thatKis convex,Lis a “layer” (i.e.,L={x: ax, ub} for somea, bRanduRn), and the centers of mass (with respect toμ) ofKandLcoincide. Thenμ(K∩L)μ(K)·μ(L). This is motivated by the well-known “positive correlation conjecture” for symmetric sets and a related inequality of Sidak concerning confidence regions for means of multivariate normal distributions. The proof uses the estimateΦ(x)> 1−((8/π)1/2/(3x+(x2+8)1/2))e−x2/2,x>−1, for the (standard) Gaussian cumulative distribution function, which is sharper than the classical inequality of Komatsu. 相似文献
19.
We consider a Cauchy problem for a semilinear heat equation with p>pS where pS is the Sobolev exponent. If u(x,t)=(T−t)−1/(p−1)φ((T−t)−1/2x) for xRN and t[0,T), where φ is a regular positive solution of(P) then u is called a backward self-similar blowup solution. It is immediate that (P) has a trivial positive solution κ≡(p−1)−1/(p−1) for all p>1. Let pL be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ for pS<p<pL. We show that there exist no radial regular positive solutions of (P) which are spatially inhomogeneous for p>pL. 相似文献
20.
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