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1.
For Ξ∈R n ,tR andfS(R n ) define $\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$ . We determine the optimal regularitys 0 such that $\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$ holds whereC is independent offS(R n ) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11]. Our theorems can be generalized to the case where the exp(it|ξ|2) is replaced by exp(it|ξ|a),a≠2. The proof uses Parseval's formula onR, orthogonality arguments arising from decomposingL 2(R n ) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments are used to show that results are sharp with respect to regularity.  相似文献   

2.
Consider the following functional equations of neutral type: $$\begin{gathered} (i) (d/dt)D(t,x_t ) = L(t,x_t ), \hfill \\ (ii) (d/dt)D(t,x_t ) = L(t,x_t ) + B(t)u(t), \hfill \\ (iii) (d/dt)D(t,x_t ) = L(t,x_t ) + B(t)u(t) + f(t,x(t),u(t)), \hfill \\ \end{gathered} $$ whereD, L are bounded linear operators fromC([?h, 0],E n) intoE n for eacht?(σ, ∞) =J, B is ann ×m continuous matrix function,u:JC m is square integrable with values in the unitm-dimensional cubeC m, andf(t, 0, 0)=0. We prove that, if the system (i) is uniformly asymptotically stable and if the controlled system (ii) is controllable, then the system (iii) is null-controllable with constraints, provided that $$f = f_1 + f_2 $$ , where $$\begin{gathered} |f_1 (t,\phi ,0)| \leqslant \varepsilon \parallel \phi \parallel , |f_2 (t,\phi ,0)| \leqslant \pi (t)\parallel \phi \parallel , t \geqslant \sigma , \hfill \\ \Pi = \int_0^\infty {\pi (t)dt< \infty .} \hfill \\ \end{gathered} $$   相似文献   

3.
A local existence and uniqueness result for the functional differential equation in a Banach space X (FDE) $$\begin{gathered} x\prime (t) \in f(t)x(t) + g(t)x_t ,0 \leqslant t \leqslant T, \hfill \\ x_0 = \phi \in C( - R,0;X) \hfill \\ \end{gathered} $$ is obtained, for the case where the operatorsf(t) satisfy only a local dissipativity condition and the operatorsg(t) are only locally Lipschitz continuous. The conditions include equations defined on cones.  相似文献   

4.
Let N denote the Hardy-Littlewood maximal operator for the familyR of one parameter rectangles. In this paper, we obtain that for 1 w p (lr) to L W P (lr) if and only if w ∈ AP(R); for 1≤p<∞, N is bounded from L W P (lr) to weak L W P (lr) if and only if W ∈ AP(R). Here we say W∈Ap (1), if $$\begin{gathered} \mathop {sup}\limits_{R \in R} \left( {\tfrac{1}{{|R|}}\smallint _r wdx} \right)\left( {\tfrac{1}{{|R|}}\smallint _R w^{ - 1/(p - 1)} dx} \right)^{p - 1}< \infty ,1< p< \infty , \hfill \\ (Nw)(x) \leqslant Cw(x)a.e.,p = 1 \hfill \\ \end{gathered} $$ ,  相似文献   

5.
Let Δ q be the set of functionsf for which theqth difference, is nonnegative on the interval [? 1,1],P n is the set of algebraic polynomials of degree not exceedingn, τ k (f, δ) p is the averaged Sendov-Popov modulus of smoothness in theL p [?1,1] metric for 1≦p≦∞, ω k (f, δ) and $\omega _\phi ^k (f,\delta ),\phi (x): = \sqrt {1 - x^2 } ,$ , are the usual modulus and the Ditzian-Totik modulus of smoothness in the uniform metric, respectively. For a functionfC[?1,1]?Δ2 we construct a polynomialp n P n 2 such that $$\begin{gathered} \left| {f(x) - p_n (x)} \right| \leqslant C\omega _3 (f,n^{ - 1} \sqrt {1 - x^2 } + n^{ - 2} ),x \in [ - 1,1]; \hfill \\ \left\| {f - p_n } \right\|_\infty \leqslant C\omega _\phi ^3 (f,n^{ - 1} ); \hfill \\ \left\| {f - p_n } \right\|_p \leqslant C\tau _3 (f,n^{ - 1} )_p . \hfill \\ \end{gathered}$$ As a consequence, for a functionfC 2[?1,1]?Δ3 a polynomialp n * P n 3 exists such that $$\left\| {f - p_n^* } \right\|_\infty \leqslant Cn^{ - 1} \omega _2 (f\prime ,n^{ - 1} ),$$ wheren≥2 andC is an absolute constant.  相似文献   

6.
The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞  相似文献   

7.
Let L: Ω × R m × R m × n R be a Caratheodory integrand with $c_1 |\nu |^{p(x)} + c_2 \leqslant L(x,u,\nu ) \leqslant c_3 |\nu |^{p(x)} + c_4 ,c_3 \geqslant c_1 > 0,n + \varepsilon \leqslant p( \cdot ) \leqslant p < \infty ,\varepsilon > 0.$ Under these assumptions the weak convergence theory holds for the integral functional $J(u): = \int\limits_\Omega {L(x,u(x),Du(x))dx} $ without further requirements. Weak convergence theory includes lower seraicontinuity with respect to the weak convergence of Sobolev functions, the convergence in energy property (weak convergence of Sobolev functions and convergence in energy imply the strong convergence of the functions), the integral representation for the relaxed energy and related questions. The results of the weak convergence theory follows from a characterization of gradient Young measures associated with these functionals.  相似文献   

8.
The paper is a supplement to [2]. LetL be a lattice andU ano-symmetric convex body inR n . The Minkowski functional? n ofU, the polar bodyU 0, the dual latticeL *, the covering radius μ(L, U), and the successive minima λ i ,i=1, …,n, are defined in the usual way. Let $\mathcal{L}_n $ be the family of all lattices inR n . Given a convex bodyU, we define $$\begin{gathered} mh(U){\text{ }} = {\text{ }}\sup {\text{ }}\max \lambda _i (L,U)\lambda _{n - i + 1} (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n 1 \leqslant i \leqslant n \hfill \\ lh(U){\text{ }} = {\text{ }}\sup {\text{ }}\lambda _1 (L,U) \cdot \mu (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n \hfill \\ \end{gathered} $$ and kh(U) is defined as the smallest positive numbers for which, given arbitrary $L \in \mathcal{L}_n $ andxR n /(L+U), somey∈L * with ∥y U 0?sd(xy,Z) can be found. It is proved $$C_1 n \leqslant jh(U) \leqslant C_2 nK(R_U^n ) \leqslant C_3 n(1 + \log n),$$ , for j=k, l, m, whereC 1,C 2,C 3 are some numerical constants andK(R U n ) is theK-convexity constant of the normed space (R n , ∥∥U). This is an essential strengthening of the bounds obtained in [2]. The bounds for lh(U) are then applied to improve the results of Kannan and Lovász [5] estimating the lattice width of a convex bodyU by the number of lattice points inU.  相似文献   

9.
We investigate the spectral singularities and the eigenvalues of the boundary value problem $$\begin{gathered} y'' + \left[ {\lambda - Q\left( x \right)} \right]^2 y = 0,x \in R_ + = [0,\infty ), \hfill \\ \quad \int\limits_0^\infty {K\left( x \right)y\left( x \right)dx + \alpha y'\left( 0 \right) - \beta y\left( 0 \right) = 0,} \hfill \\ \end{gathered}$$ where Q and K are complex valued functions, KL 2(R +), α,βC with |α|+|β|≠0 and λ is a spectral parameter.  相似文献   

10.
More work is done to study the explicit, weak and strong implicit difference solution for the first boundary problem of quasilinear parabolic system: $$\begin{gathered} u_t = ( - 1)^{M + 1} A(x,t,u, \cdots ,u_x M - 1)u_x 2M + f(x,t,u, \cdots u_x 2M - 1), \hfill \\ (x,t) \in Q_T = \left| {0< x< l,0< t \leqslant T} \right|, \hfill \\ u_x ^k (0,t) = u_x ^k (l,t) = 0 (k = 0,1, \cdots ,M - 1),0< t \leqslant T, \hfill \\ u(x,0) = \varphi (x),0 \leqslant x \leqslant l, \hfill \\ \end{gathered} $$ whereu, ?, andf arem-dimensional vector valued functions, A is anm×m positively definite matrix, and $u_t = \frac{{\partial u}}{{\partial t}},u_x ^k = \frac{{\partial ^k u}}{{\partial x^k }}$ . For this problem, the convergence of iteration for the general difference schemes is proved.  相似文献   

11.
LetZ N (t) be anN-parameter Wiener process in one dimension, and $$E(x,T) = \left\{ {t = (t_1 , \cdot \cdot \cdot ,t_N ):Z_N (t) = x,0 \leqslant t_1 , \cdot \cdot \cdot ,t_N \leqslant T} \right\}$$ . Then we obtain that with probability one, the Hausdorff measure function ofE(x,T) is $$\psi _N (r) = r^{N - \tfrac{1}{2}} (\log \log r^{ - 1} )^{\tfrac{1}{2}} ,\forall r \in (0,\frac{1}{4})$$ for anyxR 1 andT>0.  相似文献   

12.
For the functional differential equationu (n) (t)=f(u)(t) we have established the sufficient conditions for solvability and unique solvability of the boundary value problems $$u^{(i)} (0) = c_i (i = 0,...,m - 1), \smallint _0^{ + \infty } |u^{(m)} (t)|^2 dt< + \infty $$ and $$\begin{gathered} u^{(i)} (0) = c_i (i = 0),...,m - 1, \hfill \\ \smallint _0^{ + \infty } t^{2j} |u^{(j)} (t)|^2 dt< + \infty (j = 0,...,m), \hfill \\ \end{gathered} $$ wheren≥2,m is the integer part of $\tfrac{n}{2}$ ,c i R, andf is the continuous operator acting from the space of (n?1)-times continuously differentiable functions given on an interval [0,+∞] into the space of locally Lebesgue integrable functions.  相似文献   

13.
LetT be a possibly unbounded linear operator in the Banach spaceX such thatR(t)=(t+T)?1 is defined onR +. LetS=TR(I?TR) and letB(.,.) denote the Beta function. Theorem 1.1.T is a scalar-type spectral operator with spectrum in [0, ∞) if and only if $$sup\left\{ {B\left( {k,k} \right)^{ - 1} \int_0^\infty {\left| {x*S^k \left( t \right)x} \right|{{dt} \mathord{\left/ {\vphantom {{dt} t}} \right. \kern-\nulldelimiterspace} t};\left\| x \right\| \leqslant 1,} \left\| {x*} \right\| \leqslant 1,k \geqslant 1} \right\}< \infty .$$ A “local” version of this result is formulated in Theorem 2.2.  相似文献   

14.
Estimates are obtained for the nonsymmetric deviations Rn [sign x] and Rn [sign x]L of the function sign x from rational functions of degree ≤n, respectively, in the metric $$c([ - 1, - \delta ] \cup [\delta ,1]), 0< \delta< exp( - \alpha \surd \overline n ), \alpha > 0,$$ and in the metric L[?1, 1]: $$\begin{gathered} R_n [sign x] _{\frown }^\smile exp \{ - \pi ^2 n/(2 ln 1/\delta )\} , n \to \infty , \hfill \\ 10^{ - 3} n^{ - 2} \exp ( - 2\pi \surd \overline n )< R_n [sign x_{|L}< \exp ( - \pi \surd \overline {n/2} + 150). \hfill \\ \end{gathered} $$ Let 0 < δ < 1, Δ (δ)=[?1, ? δ] ∪ [δ, 1]; $$\begin{gathered} R_n [f;\Delta (\delta )] = R_n [f] = inf max |f(x) - R(x)|, \hfill \\ R_n [f;[ - 1,1] ]_L = R_n [f]_L = \mathop {inf}\limits_{R(x)} \smallint _{ - 1}^1 |f(x) - R(x)|dx, \hfill \\ \end{gathered} $$ where R(x) is a rational function of order at most n. Bulanov [1] proved that for δ ε [e?n, e?1] the inequality $$\exp \left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta }}} \right) \leqslant R_n [sign x] \leqslant 30 exp\left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta + 4 ln ln (e/\delta ) + 4}}} \right)$$ is valid. The lower estimate in this inequality was previously obtained by Gonchar ([2], cf. also [1]).  相似文献   

15.
Letf(X; T 1, ...,T n) be an irreducible polynomial overQ. LetB be the set ofb teZ n such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F j}{F j } j?1 be a Følner sequence inZ n — that is, a sequence of finite nonempty subsetsF j ?Z n such that for eachvteZ n , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP n ?A n and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ n is identified withA n (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ .  相似文献   

16.
Some estimates for simultaneous polynomial approximation of a function and its derivatives are obtained. These estimates are exact in a certain sense. In particular, the following result is derived as a corollary: Forf∈C r[?1,1],mN, and anyn≥max{m+r?1, 2r+1}, an algebraic polynomialP n of degree ≤n exists that satisfies $$\left| {f^{\left( k \right)} \left( x \right) - P_n^{\left( k \right)} \left( {f,x} \right)} \right| \leqslant C\left( {r,m} \right)\Gamma _{nrmk} \left( x \right)^{r - k} \omega ^m \left( {f^{\left( r \right)} ,\Gamma _{nrmk} \left( x \right)} \right),$$ for 0≤k≤r andx ∈ [?1,1], where ωυ(f(k),δ) denotes the usual vth modulus of smoothness off (k), and Moreover, for no 0≤k≤r can (1?x 2)( r?k+1)/(r?k+m)(1/n2)(m?1)/(r?k+m) be replaced by (1-x2)αkn2αk-2, with αk>(r-k+a)/(r-k+m).  相似文献   

17.
18.
We consider two convex polyhedra related to the vertex packing problem for a finite, undirected, loopless graphG with no multiple edges. A characterization is given for the extreme points of the polyhedron \(\mathcal{L}_G = \{ x \in R^n :Ax \leqslant 1_m ,x \geqslant 0\} \) , whereA is them × n edge-vertex incidence matrix ofG and 1 m is anm-vector of ones. A general class of facets of = convex hull{xR n :Ax≤1 m ,x binary} is described which subsumes a class examined by Padberg [13]. Some of the results for are extended to a more general class of integer polyhedra obtained from independence systems.  相似文献   

19.
I begin with a new short proof of: (I) LetP(t) inR d be a function oft havingn continuous derivatives foratx. ThenP(x)∈ convK, where $$K = \left\{ {\sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}} P^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}P^{(n)} (t),a \leqslant t \leqslant x} \right\}.$$ for applying (I) let bef(t) a real function such that the point ((t?a) n+1,f(t)) fulfills the conditions of (I). Then (I) gives a sharper estimate of then th remainder term off(x) than the Lagrange remainder formula. Iff( n )(t) is also convex inatx, thenf(x)∈[c,d], where $$\begin{gathered} c = \sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}f^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}f^{(n)} \left( {\frac{{na + x}}{{n + 1}}} \right)} , \hfill \\ d = \sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}f^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}} \frac{{nf^{(n)} (a) + f^{(n)} (x)}}{{n + 1}}. \hfill \\ \end{gathered} $$   相似文献   

20.
We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x 2)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S n ?1 (f) = S n ?1 (f, x) is the projection onto the subspace of algebraic polynomials p n = p n (x) of degree at most n, i.e., S n (p n ) = p n ; in addition, S n ?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S n ?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ n (x) of the partial sums S n ?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .  相似文献   

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