共查询到20条相似文献,搜索用时 15 毫秒
1.
Denote by B
2σ,p
(1 < p < ∞) the bandlimited class p-integrable functions whose Fourier transform is supported in the interval [−σ, σ]. It is shown that a function in B
2σ,p
can be reconstructed in L
p(ℝ) by its sampling sequences {f (κπ / σ)}
κ∈ℤ and {f’ (κπ / σ)}
κ∈ℤ using the Hermite cardinal interpolation. Moreover, it will be shown that if f belongs to L
p
r
(ℝ), 1 < p < ∞, then the exact order of its aliasing error can be determined.
Project supported by the Scientific Research Common Program of Beijing Municipal Commission of Education under grant number
KM 200410009010 and by the Natural Science Foundation of China under grant number 10071006 相似文献
2.
Hans TRIEBEL 《数学学报(英文版)》2008,24(4):539-554
A space Apq^s (R^n) with A : B or A = F and s ∈R, 0 〈 p, q 〈 ∞ either has a trace in Lp(Г), where Г is a compact d-set in R^n with 0 〈 d 〈 n, or D(R^n/Г) is dense in it. Related dichotomy numbers are introduced and calculated. 相似文献
3.
Soulaymane Korry 《Israel Journal of Mathematics》2003,133(1):357-367
Letp∈(1, +∞) ands ∈ (0, +∞) be two real numbers, and letH
p
s
(ℝ
n
) denote the Sobolev space defined with Bessel potentials. We give a classA of operators, such thatB
s,p
-almost all points ℝ
n
are Lebesgue points ofT(f), for allf ∈H
p
s
(ℝ
n
) and allT ∈A (B
s,p
denotes the Bessel capacity); this extends the result of Bagby and Ziemer (cf. [2], [15]) and Bojarski-Hajlasz [4], valid
wheneverT is the identity operator. Furthermore, we describe an interesting special subclassC ofA (C contains the Hardy-Littlewood maximal operator, Littlewood-Paley square functions and the absolute value operatorT: f→|f|) such that, for everyf ∈H
p
s
(ℝ
n
) and everyT ∈C, T(f) is quasiuniformly continuous in ℝ
n
; this yields an improvement of the Meyers result [10] which asserts that everyf ∈H
p
s
(ℝ
n
) is quasicontinuous. However,T (f) does not belong, in general, toH
p
s
(ℝ
n
) wheneverT ∈C ands≥1+1/p (cf. Bourdaud-Kateb [5] or Korry [7]). 相似文献
4.
Let Ω be a bounded Lipschitz domain. Define B
0,1
1,
r
(Ω) = {f∈L
1 (Ω): there is an F∈B
0,1
1 (ℝ
n
) such that F|Ω = f} and B
0,1
1
z
(Ω) = {f∈B
0,1
1 (ℝ
n
) : f = 0 on ℝ
n
\}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the
regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation of ℝ
n
+.
Received June 8, 2000, Accepted October 24, 2000 相似文献
5.
Daniel Berend 《Journal d'Analyse Mathématique》1985,45(1):255-284
The study of jointly ergodic measure preserving transformations of probability spaces, begun in [1], is continued, and notions
of joint weak and strong mixing are introduced. Various properties of ergodic and mixing transformations are shown to admit
analogues for several transformations. The case of endomorphisms of compact abelian groups is particularly emphasized. The
main result is that, given such commuting endomorphisms σ1σ2,...,σ, ofG, the sequence ((1/N)Σ
n=0
N−1
σ
1
n
f
1·σ
2
n
f
2· ··· · σ
s
n
f
sconverges inL
2(G) for everyf
1,f
2,…,f
s∈L
∞(G). If, moreover, the endomorphisms are jointly ergodic, i.e., if the limit of any sequence as above is Π
i=1
s
∫
G
f
1
d
μ, where μ is the Haar measure, then the convergence holds also μ-a.e. 相似文献
6.
Jerome A. Goldstein 《Semigroup Forum》1996,52(1):37-47
Of concern are semigroups of linear norm one operators on Hilbert space of the form (discrete case)T={T
n
/n=0,1,2,...} or (continuous case)T={T(t)/t=≥0}. Using ergodic theory and Hilbert-Schmidt operators, the Cesàro limits (asn→∞) of |〈T
n
f,f〉|2, |〈T
(n)f,f〉|2 are computed (withn∈ℤ+ orn∈ℤ+). Specializing the Hilbert space to beL
2(T,μ) (discrete case) orL
2(ℝ,μ) (continuous case) where μ is a Borel probability measure on the circle group or the line, the Cesàro limit of
(asn→±∞, with,n∈ℤ orn∈ℝ) is obtained and interpreted. Extensions toT
M
, and ℝ
M
are given. Finally, we discuss recent operator theoretic extensions from a Hilbert to a Banach space context.
Partially supported by an NSF grant 相似文献
7.
N. V. Lazakovich S. P. Stashulenok O. L. Yablonskii 《Lithuanian Mathematical Journal》1999,39(2):196-202
In this paper, we consider problems of approximation of stochastic θ-integrals (θ)∫
0
t
f(B(s))dB(s) with respect to a Brownian motion by sums of the form ∑
k=1
p
fn(B
n
θ
(tk-1))[B
n
θ
(tk)-B
n
θ
(tk-1], where the sequences {fn,n∈∕#x007D; and {[B
n
θ
,n∈∕} are convolution-type approximations of the functionf and Brownian motionB.
Belorussian State University, F. Skoryna ave. 4, 220050 Minsk, Belorus. Translated from Lietuvos Matematikos Rinkinys, Vol.
39, No. 2, pp. 248–256, April–June, 1999.
Translated by V. Mackevičius 相似文献
8.
Fang Liping 《数学学报(英文版)》1998,14(1):139-144
Letf be a holomorphic self-map of the punctured plane ℂ*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI
0(f)={z ∈ ℂ*:f
n
(z) → 0,n → ∞} andI
∞(f)={z ∈ ℂ*:f
n
(z) → 0,n → ∞}. We try to find the relation betweenI
0(f),I
∞(t) andJ(f). It is proved that both the boundary ofI
0(f) and the boundary ofI
∞)f) equal toJ(f),I
0(f) ∩J(f) ≠ θ andI
∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI
0(f) andI
∞(f) are not doubly-bounded.
Supported by the National Natural Science Foundation of China 相似文献
9.
Liguang Liu 《Frontiers of Mathematics in China》2007,2(4):599-611
Let ℐ(ℝn) be the Schwartz class on ℝn and ℐ∞(ℝn) be the collection of functions ϕ ∊ ℐ(ℝn) with additional property that
for all multiindices γ. Let (ℐ(ℝn))′ and (ℐ∞(ℝn))′ be their dual spaces, respectively. In this paper, it is proved that atomic Hardy spaces defined via (ℐ(ℝn))′ and (ℐ∞(ℝn))′ coincide with each other in some sense. As an application, we show that under the condition that the Littlewood-Paley
function of f belongs to L
p(ℝn) for some p ∊ (0,1], the condition f ∊ (ℐ∞(ℝn))′ is equivalent to that f ∊ (ℐ(ℝn))′ and f vanishes weakly at infinity. We further discuss some new classes of distributions defined via ℐ(ℝn) and ℐ∞(ℝn), also including their corresponding Hardy spaces.
相似文献
10.
S. Norvidas 《Lithuanian Mathematical Journal》2009,49(2):185-189
For a compact set K in ℝ
n
, let B
2
K
be the set of all functions f ∈ L
2(ℝ2) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B
2
K
by finite exponential sums
in the space , as τ → ∞. 相似文献
11.
The spectrum of each symmetric ψ DO of the symbol class S0
1, γ, 0≤γ<1, acting on B3
p,q(w(x)) and F3
p,q(w(x)), is independent of the choice ofs ∈ℝ, 0<p≤∞ (p<∞ in the F-case), 0<q≤∞ and the weight w(x)∈W. 相似文献
12.
Integration questions related to fractional Brownian motion 总被引:1,自引:0,他引:1
Let {B
H
(u)}
u
∈ℝ be a fractional Brownian motion (fBm) with index H∈(0, 1) and (B
H
) be the closure in L
2(Ω) of the span Sp(B
H
) of the increments of fBm B
H
. It is well-known that, when B
H
= B
1/2 is the usual Brownian motion (Bm), an element X∈(B
1/2) can be characterized by a unique function f
X
∈L
2(ℝ), in which case one writes X in an integral form as X = ∫ℝ
f
X
(u)dB
1/2(u). From a different, though equivalent, perspective, the space L
2(ℝ) forms a class of integrands for the integral on the real line with respect to Bm B
1/2. In this work we explore whether a similar characterization of elements of (B
H
) can be obtained when H∈ (0, 1/2) or H∈ (1/2, 1). Since it is natural to define the integral of an elementary function f = ∑
k
=1
n
f
k
1
[uk,uk+1)
by ∑
k
=1
n
f
k
(B
H
(u
k
+1) −B
H
(u
k
)), we want the spaces of integrands to contain elementary functions. These classes of integrands are inner product spaces.
If the space of integrands is not complete, then it characterizes only a strict subset of (B
H
). When 0<H<1/2, by using the moving average representation of fBm B
H
, we construct a complete space of integrands. When 1/2<H<1, however, an analogous construction leads to a space of integrands which is not complete. When 0<H<1/2 or 1/2<H<1, we also consider a number of other spaces of integrands. While smaller and henceincomplete, they form a natural choice
and are convenient to workwith. We compare these spaces of integrands to the reproducing kernel Hilbert space of fBm.
Received: 9 August 1999 / Revised version: 10 January 2000 / Published online: 18 September 2000 相似文献
13.
Let L
p
(S), 0 < p < +∞, be a Lebesgue space of measurable functions on S with ordinary quasinorm ∥·∥
p
. For a system of sets {B
t
|t ∈ [0, +∞)
n
} and a given function ψ: [0, +∞)
n
↦ [ 0, +∞), we establish necessary and sufficient conditions for the existence of a function f ∈ L
p
(S) such that inf {∥f − g∥
p
p
g ∈ L
p
(S), g = 0 almost everywhere on S\B
t
} = ψ (t), t ∈ [0, +∞)
n
. As a consequence, we obtain a generalization and improvement of the Dzhrbashyan theorem on the inverse problem of approximation
by functions of the exponential type in L
2.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1116–1127, August, 2006. 相似文献
14.
WU Hao & LI Weigu School of Mathematical Sciences Peking University Beijing China 《中国科学A辑(英文版)》2005,48(12):1670-1682
In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case. 相似文献
15.
The purpose of this paper is to study the L
2 boundedness of operators of the form f ↦ ψ(x) ∫ f (γ
t
(x))K(t)dt, where γ
t
(x) is a C
∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝ
N
× ℝ
n
, satisfying γ
0(x) ≡ x, ψ is a C
∞ cut-off function supported on a small neighborhood of 0 ∈ ℝ
n
, and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝ
N
. The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L
2. The case when K is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case
when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderón- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later
two of which are joint with E. M. Stein. The second paper deals with the related question of L
p
boundedness, while the third paper deals with the special case when γ is real analytic. 相似文献
16.
17.
F. V. Petrov 《Journal of Mathematical Sciences》2007,147(6):7218-7226
Let Γ ⊂ ℝd be a bounded strictly convex surface. We prove that the number kn(Γ) of points of Γ that lie on the lattice
satisfies the following estimates: lim inf kn(Γ)/nd−2 < ∞ for d ≥ 3 and lim inf kn(Γ)/log n < ∞ for d = 2. Bibliography: 9 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 344, 2007, pp. 174–189. 相似文献
18.
Bogus?aw Bo?ek Wies?aw Solak Zbigniew Szyde?ko 《Central European Journal of Mathematics》2012,10(3):1172-1184
We investigate quadrature rules with Laplace end corrections that depend on a parameter β. Specific values of β yield sixth order rules. We apply our results to approximating the sum of slowly converging series s = Σ
i=1∞
f(i + 1/2) where f ∈ C
6 with its sixth derivative of constant sign on [m, ∞) and ∫
m
∞
f(x)dx is known for m ∈ ℕ. Several examples show the efficiency of this method. This paper continues the results from [Solak W., Szydełko Z., Quadrature
rules with Gregory-Laplace end corrections, J. Comput. Appl. Math., 1991, 36(2), 251–253]. 相似文献
19.
Gerard van der Laan 《Mathematical Programming》1984,28(1):1-24
In this paper we consider the problem of finding zeroes of a continuous functionf from a convex, compact subsetU of ℝ
n
to ℝ
n
. In the first part of the paper it is proved thatf has a computable zero iff:C
n
→ℝ
n
satisfies the nonparallel condition for any two antipodal points on bdC
n, i.e. if for anyx∈bdC
n
,f(x)≠αf(−x), α≥0, holds. Therefore we describe a simplicial algorithm to approximate such a zero. It is shown that generally the degree
of the approximate zero depends on the number of reflection steps made by the algorithm, i.e. the number of times the algorithm
switches from a face τ on bdC
n
to the face −τ. Therefore the index of a terminal simplex σ is defined which equals the local Brouwer degree of the function
if σ is full-dimensional. In the second part of the paper the algorithm is used to generate possibly several approximate zeroes
off. Two sucessive solutions may have both the same or opposite degrees, again depending on the number of reflection steps. By
extendingf:U→ℝ
n
to a function g from a cube containingU to ℝ
n
, the procedure can be applied to any continuous functionf without having any information about the global and local Brouwer degrees a priori. 相似文献
20.
Michael T. Lacey 《Journal d'Analyse Mathématique》1995,67(1):199-206
LetT
1 andT
2 be commuting invertible ergodic measure preserving flows on a probability space (X, A, μ). For t = (u,ν) ∈ ℝ2, letT
t
=T
1
u
T
2
v
. LetS
1 denote the unit circle in ℝ2 and σ the rotation invariant unit measure on it. Then, forf∈Lp(X) withp>2, the averagesA
t
f(x) = ∫
s
1
f(T
ts
x)σ(ds) conver the integral off for a. e.x, ast tends to 0 or infinity. This extends a result of R. Jones [J], who treated the case of three or more dimensions. 相似文献