共查询到20条相似文献,搜索用时 15 毫秒
1.
Thomas Kuhn Hans-Gerd Leopold Winfried Sickel Leszek Skrzypczak 《Constructive Approximation》2005,23(1):61-77
We investigate the asymptotic behavior of the entropy numbers of the
compact embedding
$$
B^{s_1}_{p_1,q_1} \!\!(\mbox{\footnotesize\bf R}^d, \alpha) \hookrightarrow B^{s_2}_{p_2,q_2} \!\!({\xxR}).
$$
Here $B^s_{p,q} \!({\mbox{\footnotesize\bf R}^d}, \alpha)$ denotes a weighted Besov space, where the weight is
given by $w_\alpha (x) = (1+| x |^2)^{\alpha/2}$, and
$B^{s_2}_{p_2,q_2} \!({\mbox{\footnotesize\bf R}^d})$ denotes the unweighted Besov space, respectively.
We shall concentrate
on the so-called limiting situation given by the following constellation of
parameters: $s_2 < s_1$, $0 < p_1,p_2 \le \infty$, and
$$
\alpha = s_1 - \frac{d}{p_1} - s_2 + \frac{d}{p_2} >
d \, \max \Big(0, \frac{1}{p_2}-\frac{1}{p_1}\Big).
$$
In almost all cases we give a sharp two-sided estimate. 相似文献
2.
赵俊燕 《数学年刊A辑(中文版)》2017,38(4):405-418
研究了欧氏空间R~2中单位方体Q~2=[0,1]~2上沿曲面(t,s,γ(t,s))的振荡奇异积分算子T_(α,β)f(u,v,x)=∫_(Q~2)f(u-t,v-s,x-γ(t,s))e~(it~(-β_1)s~(-β_2))t~(-1-α_1)s~(-1-α_2)dtds从Sobolev空间L_τ~p(R~(2+n))到L~p(R~(2+n))中的有界性,其中x∈R~n,(u,v)∈R~2,(t,s,γ(t,s))=(t,s,t~(P_1)s~(q_1),t~(p_2)s~(q_2),…,t~(p_n)s~(q_n))为R~(2+n)上一个曲面,且β_1α_1≥0,β_2α_20.这些结果推广和改进了R~3上的某些已知的结果.作为应用,得到了乘积空间上粗糖核奇异积分算子的Sobolev有界性. 相似文献
3.
设$W_{\beta}(x)=\exp(-\frac{1}{2}|x|^{\beta})~(\beta > 7/6)$ 为Freud权, Freud正交多项式定义为满足下式$\int_{- \infty}^{\infty}p_{n}(x)p_{m}(x)W_{\beta}^{2}(x)\rd x=\left \{ \begin{array}{ll} 0 & \hspace{3mm} n \neq m , \\ 1 & \hspace{3mm}n = m \end{array} \right.$的 相似文献
4.
V. Maiorov 《Constructive Approximation》2005,21(3):337-349
We investigate the best approximation of some
classes of functions defined on a $d$-dimensional torus
${\mbox{\smallbf T}}^d$ by the transfer manifold
$H_n(\varphi)=\{\xxsum_{i=1}^n b_i\varphi(\cdot -a_i)\}$
in the Hilbert space $L_2({\mbox{\smallbf T}}^d)$. For Sobolev classes of
functions $\tilde{W}_2^r$ we obtain two-sided estimates of the deviation
${\rm dist}(\tilde{W}_2^r,H_n(\varphi))$. These results are
applied to obtain estimates for radial basis approximation. 相似文献
5.
We develop structural formulas
satisfied by some families of
orthogonal matrix polynomials of size $2\times 2$ satisfying
second-order differential equations with polynomial coefficients. We consider
here two one-parametric families of weight matrices,
namely
\[
H_{a,1}(t)\;=\;e^{-t^2} \left( \begin{array}{@{}cc@{}}
1+\vert a\vert ^2t^2 & at \\bar at & 1 \end{array} \right) \quad {\rm and} \quad H_{a,2}(t)\;=\;e^{-t^2} \left( \begin{array} {@{}cc@{}}
1+\vert a\vert ^2t^4 & at^2 \\bar at^2 & 1
\end{array} \right),
\]
$a\in \mbox{\bf C} $ and $t\in \mbox{\bf R} $, and their corresponding orthogonal
polynomials. 相似文献
6.
Let H be a Hopf π-coalgebra over a commutative ring k with bijective antipode S, and A and B right π-H-comodulelike algebras. We show that the pair of adjoint functors (F3 = A Bop A□ HBop -,G3 = (-)coH) between the categories A□HBopM and AMπB-H is a pair of inverse equivalences, when A is a faithfully flat π-H-Galois extension. Furthermore, the categories Moritaπ-H(A,B) and Morita □π-H(AcoH,BcoH) are equivalent, if A and B are faithfully flat π-H-Galois extensions. 相似文献
7.
假设a,b0并且K_(a,b)(x)=(e~(i|x|~(-b)))/(|x|~(n+a))定义强奇异卷积算子T如下:Tf(x)=(K_(a,b)*f)(x),本文主要考虑了如上定义的算子T在Wiener共合空间W(FL~p,L~q)(R~n)上的有界性.另一方面,设α,β0并且γ(t)=|t|~k或γ(t)=sgn(t)|t|~k.利用振荡积分估计,本文还研究了算子T_(α,β)f(x,y)=p.v∫_(-1)~1f(x-t,y-γ(t))(e~(2πi|t|~(-β)))/(t|t|~α)dt及其推广形式∧_(α,β)f(x,y,z)=∫_(Q~2)f(x-t,y-s,z-t~ks~j)e~(-2πit)~(-β_1_s-β_2)t~(-α_1-1)s~(-α_2-1)dtds在Wiener共合空间W(FL~p,L~q)上的映射性质.本文的结论足以表明,Wiener共合空间是Lebesgue空间的一个很好的替代. 相似文献
8.
刘桥 《数学年刊A辑(中文版)》2014,35(5):591-612
考虑了R~n上n(n≥2)维向列型液晶流(u,d)当初值属于Q_α~(-1)(R~n,R~n)×Q_α(R~n,S~2)(其中α∈(0,1))时Cauchy问题的适定性,这里的Q_α(R~n)最早由Essen,Janson,Peng和Xiao(见[Essen M,Janson S,Peng L,Xiao J.Q space of several real variables,Indiana Univ Math J,2000,49:575-615])引入,是指由R~n中满足的所有可测函数f全体所组成的空间.上式左端在取遍Rn中所有以l(I)为边长且边平行于坐标轴的立方体I的全体中取上确界,而Q_α~(-1)(R~n):=▽·Q_α(R~n).最后证明了解(u,d)在类C([0,T);Q_(α,T)~(-1)(R~n,R~n))∩L_(loc)~∞((0,T);L~∞(R~n,R~n))×C([0,T);Q_α,T(R~n,S~2))∩L_(loc)~∞((0,T);W~(1,∞)(R~n,S~2))(其中0T≤∞)中是唯一的. 相似文献
9.
Generalized Shift-Invariant Systems 总被引:1,自引:0,他引:1
A countable collection $X$ of functions in $L_2(\mbox{\footnotesize\bf
R})$ is said to be a Bessel system if the associated
analysis operator
$$
\txs{X}:L_2(\mbox{\smallbf R}^d)\to \ell_2(X) : f\mapsto (\inpro{f,x})_{x\in X}
$$
is well-defined and bounded. A Bessel system is a fundamental frame if
$\txs{X}$ is injective and its range is closed.
This paper considers the above two properties for a generalized
shift-invariant system $X$. By definition, such a system has the form
$$
X=\bigcup_{j\in J} Y_j,
$$
where each $Y_j$ is a shift-invariant system (i.e., is comprised of lattice
translates of some function(s)) and $J$ is a countable (or finite) index set.
The definition is general enough to include wavelet systems, shift-invariant
systems, Gabor systems, and many variations of wavelet systems such as
quasi-affine ones and nonstationary ones.
The main theme of this paper is the fiberization of $\txs{X}$,
which allows one to study the frame and Bessel properties of $X$ via the spectral
properties of a collection of finite-order Hermitian nonnegative matrices. 相似文献
10.
Guoen HU 《数学年刊B辑(英文版)》2017,38(3):795-814
Let T_σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that sup κ∈Z∥σ_κ∥W~s(R~(2n)) ∞ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T_σ and CMO(R~n) functions is a compact operator from L~(p1)(R~n, w_1) × L~(p2)(R~n, w_2) to L~p(R~n, ν_w) for appropriate indices p_1, p_2, p ∈ (1, ∞) with1 p=1/ p_1 +1/ p_2 and weights w_1, w_2 such that w = (w_1, w_2) ∈ A_(p/t)(R~(2n)). 相似文献
11.
N. Kruglyak E. A. Kuznetsov 《Proceedings of the American Mathematical Society》2007,135(10):3283-3293
The Marcinkiewicz integral plays a well-known and prominent role in harmonic analysis. In this paper, we estimate the growth of it in the limiting case . Throughout, we assume that is convex; it is interesting that this condition cannot be dropped.
12.
Herbert Stahl 《Constructive Approximation》2006,23(2):121-164
The asymptotic distributions of zeros of the quadratic Hermite--Pad\'{e}
polynomials $p_{n},q_{n},r_{n}\in{\cal P}_{n}$ associated with the exponential function are studied for $n\rightarrow\infty$.
The polynomials are defined by the relation
$$(*)\qquad p_{n}(z)+q_{n}(z)e^{z}+r_{n}(z)e^{2z}=O(z^{3n+2})\qquad\mbox{as} \quad z\rightarrow0,$$
and they form the basis for quadratic Hermite--Pad\'{e} approximants to $e^{z}$. In order to achieve a differentiated picture
of the asymptotic behavior of the zeros, the independent variable $z$ is rescaled in such a way that all zeros of the polynomials
$p_{n},q_{n},r_{n}$ have finite cluster points as $n\rightarrow\infty$. The asymptotic relations, which are proved, have a
precision that is high enough to distinguish the positions of individual zeros. In addition to the zeros of the polynomials
$p_{n},q_{n},r_{n}$, also the zeros of the remainder term of (*) are studied. The investigations complement asymptotic results
obtained in [17]. 相似文献
13.
Nitsche证明了共形双曲度量的孤立奇点要么是锥奇点,要么是尖奇点,二者必其一(Nitsche J.über die isolierten singularit?ten der L?sungen von△u=e~u [J]. Math Z, 1957, 68(3):316-324.).本文利用展开映射证明了在孤立奇点附近存在复坐标z,使得度量要么为(4α~2|z|~(2α-2))/((1-|z|~(2α))~2)|dz|~2,其中α 0,要么为|z|~(-2)(ln|z|)~(-2)|dz|~2. 相似文献
14.
Let B R~n be the unit ball centered at the origin. The authors consider the following biharmonic equation:{?~2u = λ(1 + u)~p in B,u =?u/?ν= 0 on ?B, where p n+4/ n-4and ν is the outward unit normal vector. It is well-known that there exists a λ* 0 such that the biharmonic equation has a solution for λ∈ (0, λ*) and has a unique weak solution u*with parameter λ = λ*, called the extremal solution. It is proved that u* is singular when n ≥ 13 for p large enough and satisfies u*≤ r~(-4/ (p-1)) - 1 on the unit ball, which actually solve a part of the open problem left in [D`avila, J., Flores, I., Guerra, I., Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348(1), 2009, 143–193] . 相似文献
15.
Cesar Enrique Torres Ledesm Ziheng Zhang Amado Mendez 《Journal of Applied Analysis & Computation》2019,9(6):2436-2453
We study the existence of solutions for the following fractional Hamiltonian systems
$$
\left\{
\begin{array}{ll}
- _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm]
u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n),
\end{array}
\right.
~~~~~~~~~~~~~~~~~(FHS)_\lambda
$$
where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $\lambda>0$ is a parameter, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric matrix, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$. Assuming that
$L(t)$ is a positive semi-definite symmetric matrix, that is, $L(t)\equiv 0$ is allowed to occur in some finite interval $T$ of $\mathbb{R}$,
$W(t,u)$ satisfies some superquadratic conditions weaker than Ambrosetti-Rabinowitz condition, we show that (FHS)$_\lambda$ has a solution which vanishes on
$\mathbb{R}\setminus T$ as $\lambda \to \infty$, and converges to some $\tilde{u}\in H^{\alpha}(\R, \R^n)$. Here, $\tilde{u}\in E_{0}^{\alpha}$ is a solution
of the Dirichlet BVP for fractional systems on the finite interval $T$. Our results are new and improve recent results in the literature even in the case $\alpha =1$. 相似文献
16.
Peter Borwein Richard Lockhart 《Proceedings of the American Mathematical Society》2001,129(5):1463-1472
The results of this paper concern the expected norm of random polynomials on the boundary of the unit disc (equivalently of random trigonometric polynomials on the interval ). Specifically, for a random polynomial
let
Assume the random variables , are independent and identically distributed, have mean 0, variance equal to 1 and, if 2$">, a finite moment . Then
and
as .
In particular if the polynomials in question have coefficients in the set (a much studied class of polynomials), then we can compute the expected norms of the polynomials and their derivatives
and
This complements results of Fielding in the case, Newman and Byrnes in the case, and Littlewood et al. in the case.
17.
洪勇 《数学年刊A辑(中文版)》2011,32(5):599-606
设核函数K(u,v)具有对称性和齐次性,对如下定义的奇异重积分算子T:(Tf)(y)=∫R_+~n K(‖x‖α,‖y‖α)f(x)dx,y∈R_+~n,其中‖x‖α=(x_1~α+…+x_n~α)~1/α(α>0),研究了T的范数及其应用. 相似文献
18.
Georg Schneider 《Proceedings of the American Mathematical Society》2004,132(8):2399-2409
We consider Hankel operators of the form . Here . We show that in the case of one complex dimension the Hankel operators are compact but not Hilbert-Schmidt if 2k$">.
19.
Let
denote the linear space over
spanned by
. Define the (real) inner product
, where V satisfies: (i) V is real analytic on
; (ii)
; and (iii)
. Orthogonalisation of the (ordered) base
with respect to
yields the even degree and odd degree orthonormal Laurent polynomials
, and
. Define the even degree and odd degree monic orthogonal Laurent polynomials:
and
. Asymptotics in the double-scaling limit
such that
of
(in the entire complex plane),
, and
(in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a
matrix Riemann-Hilbert problem on
, and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further
developed in [2],[3]. 相似文献
20.
Consider the Kirchhoff type equation \begin{equation}\label{eq0.1}-\left(a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\,dx\right) \Delta u=\left(\frac{1}{|x|^\mu}*F(u)\right)f(u)\ \ \mbox{in}\ \mathbb{R}^N, \ \ u\in D^{1,2}(\mathbb{R}^N), ~~~~~~(0.1)\end{equation}where $a>0$, $b\geq0$, $0<\mu<\min\{N, 4\}$ with $N\geq 3$, $f: \mathbb{R}\to\mathbb{R}$ is a continuous function and $F(u)=\int_0^u f(t)\,dt$. Under some general assumptions on $f$, we establish the existence of a nontrivial spherically symmetric solution for problem (0.1). The proof is mainly based on mountain pass approach and a scaling technique introduced by Jeanjean. 相似文献