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1.
Darko Zubrinić 《Archiv der Mathematik》2006,87(2):154-162
Assuming that 0 < α p < N, p, q ∈(1,∞), we construct a class of functions in the Besov space
such that the Hausdorff dimension of their singular set is equal to N − α p. We show that these functions are maximally singular, that is, the Hausdorff dimension of the singular set of any other Besov
function in
is ≦ N − α p. Similar results are obtained for Lizorkin-Triebel spaces
and for the Hardy space
. Some open problems are listed.
Received: 5 July 2005; revised: 18 October 2005 相似文献
2.
A. I. Kozko 《Journal of Mathematical Sciences》2006,139(6):7151-7164
We study the problem on the completeness of orthogonal systems in asymmetric spaces with sign-sensitive weight. Theorems of
general form are obtained. In particular, the necessary and sufficient conditions on α, β, q
1, and q
2 for which the known orthogonal systems are everywhere dense in asymmetric spaces L
(α,β);q ([0, 1]) are found.
Theorem. Let α, β, q
1, q
2 ∈ [1,+∞]. The following orthogonal systems are dense in asymmetric spaces L
(α,β);q ([0, 1]) if and only if either max{α, β, q
1, q
2} < + ∞ or max {α, β} < +∞, q
1 = q
2 = +∞: trigonometric, algebraic, Haar’s system, Meyer’s system of wavelets, Shannon-Kotel’nikov wavelets, Stromberg and Lemarie-Battle
wavelets, the Walsh system, and the Franklin system.
__________
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical
Systems and Optimization, 2005. 相似文献
3.
Abstract
In this paper, we establish the relationship between
Hausdorff measures and Bessel capacities on any nilpotent
stratified Lie group
(i. e., Carnot group). In particular, as a special corollary of
our much more general results, we have the following theorem
(see Theorems A and E in Section 1):
Let Q be the
homogeneous dimension of
.
Given any set E ⊂
,
B
α,p
(E) = 0 implies ℋ
Q−αp+ ε(E) = 0 for all ε > 0. On the other
hand, ℋ
Q−αp
(E) < ∞ implies
B
α,p
(E) = 0. Consequently, given any set
E ⊂
of Hausdorff dimension Q −
d, where 0 <
d <
Q, B
α,p
(E) = 0 holds if and only if αp ≤ d.
A version of O. Frostman’s theorem concerning Hausdorff
measures on any homogeneous space is also established using the
dyadic decomposition on such a space (see Theorem 4.4 in Section
4).
Research supported partly by the U. S. National
Science Foundation Grant No. DMS99–70352 相似文献
4.
For q ≥ 0, Olsen [1] has attained the exact rate of convergence of the L
q
-spectrum of a self-similar measure and showed that the so-called empirical multifractal moment measures converges weakly
to the normalized multifractal measures. Unfortunately, nothing is known for q < 0. Indeed, the problem of analysing the L
q
- spectrum for q < 0 is generally considered significantly more difficult since the L
q
-spectrum is extremely sensitive to small variations of μ for q < 0. In [2] we showed that self-similar measures satisfying the Open Set Condition (OSC) are Ahlfors regular and, using this
fact, we obtained the exact rate of convergence of the L
q
-spectrum of a self-similar measure satisfying the OSC for q < 0. In this paper, we apply the results from [2] to show the empirical multifractal q’th moment measures of self-similar
measures satisfying the OSC converges weakly to the normalized multifractal Hausdorff measures for q < 0.
Authors’ addresses: Jiaqing Xiao, School of Science, Wuhan University of Technology, Wuhan 430070, China; Wu Min, School of
Mathematical Sciences, South China University of Technology, Guangzhou, 510640, China 相似文献
5.
M. I. Gvaradze 《Mathematical Notes》1977,21(2):79-84
The spacesb (p, q, λ) (0<p<q⩽∞, 0<λ⩽∞) of functions, analytic in the circle |z|< 1, are introduced, and an unimprovable estimate is obtained for the Taylor coefficients
of a functionf∃
b (p, q, λ). It is shown that B(p, q, λ) is the space of fractional derivatives f(α) of order α (−∞<α<1/p−1/q) of a function
f of B(s, q, λ), where s=p/(1−αp).
Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 141–150, February, 1977. 相似文献
6.
For q ≥ 0, Olsen [1] has attained the exact rate of convergence of the L
q
-spectrum of a self-similar measure and showed that the so-called empirical multifractal moment measures converges weakly
to the normalized multifractal measures. Unfortunately, nothing is known for q < 0. Indeed, the problem of analysing the L
q
- spectrum for q < 0 is generally considered significantly more difficult since the L
q
-spectrum is extremely sensitive to small variations of μ for q < 0. In [2] we showed that self-similar measures satisfying the Open Set Condition (OSC) are Ahlfors regular and, using this
fact, we obtained the exact rate of convergence of the L
q
-spectrum of a self-similar measure satisfying the OSC for q < 0. In this paper, we apply the results from [2] to show the empirical multifractal q’th moment measures of self-similar
measures satisfying the OSC converges weakly to the normalized multifractal Hausdorff measures for q < 0. 相似文献
7.
Manuel A. Fugarolas 《Czechoslovak Mathematical Journal》2011,61(1):209-212
Let 1 ⩽ q < p < ∞ and 1/r:= 1/p max(q/2, 1). We prove that L
r,p
(c), the ideal of operators of Gel’fand type l
r,p
, is contained in the ideal Π
p,q
of (p, q)-absolutely summing operators. For q > 2 this generalizes a result of G. Bennett given for operators on a Hilbert space. 相似文献
8.
S. S. Volosivets 《P-Adic Numbers, Ultrametric Analysis, and Applications》2011,3(2):149-156
For Hausdorff operator with generating function having support in the unit ball of p-adic field ℚ
p
we give sufficient and necessary conditions of its boundedness in BMO-type spaces: BLO(ℚ
p
n
), Q
r
α,q
(ℚ
p
n
) and BMO
r
α,q
(ℚ
p
n
). Some embedding relations between these spaces and Besov spaces are established. 相似文献
9.
We generalize a method developed by Sarig to obtain polynomial lower bounds for correlation functions for maps with a countable
Markov partition. A consequence is that LS Young’s estimates on towers are always optimal. Moreover, we show that, for functions
with zero average, the decay rate is better, gaining a factor 1/n. This implies a Central Limit Theorem in contexts where it was not expected, e.g.,x+Cx
1+α with 1/2⩽α<1. The method is based on a general result on renewal sequences of operators, and gives an asymptotic estimate up to any
precision of such operators. 相似文献
10.
Fang Gensun 《中国科学A辑(英文版)》2001,44(9):1126-1131
The main result of this paper asserts that if a function f is in the class Bπ,p, 1 <p < ∞; that is, those p-integrable functions whose Fourier transforms are supported in the interval [ - π, π], then f and its
derivatives f(j) j = 1, 2, …, can be recovered from its sampling sequence{f(k)} via the cardinal interpolating spline of degree m in the metric
ofL
q(ℝ)), 1 <p=q < ∞, or 11 <p=q < ⩽ ∞. 相似文献
11.
12.
For any given 0 〈α 〈 β 〈 ∞, we construct a tree such that under tree metric, the Hausdorff dimension of the corresponding boundary is α, but both the Packing dimension and the boxing dimension are β. Applying the connection between tree and iterated functions system, non- regular fractal sets on real line are constructed. Moreover, the method adopted in our paper is different from those which have been used before for constructing non-regular fractal set in general metric space. 相似文献
13.
S. A. Stasyuk 《Ukrainian Mathematical Journal》2010,62(1):114-122
We obtain an exact-order estimate for the best m-term trigonometric approximation of the Besov classes
Bp,\uptheta r B_{p,{{\uptheta }}}^r of periodic functions of many variables of low smoothness in the space L
q
, 1 < p ≤ 2 < q < ∞. 相似文献
14.
A. S. Romanyuk 《Ukrainian Mathematical Journal》1998,50(8):1242-1252
We obtain estimates exact in order for the trigonometric widths of the Besov classes B
p,θr of periodic functions of many variables in the space L
q for 1 ≤ p ≤ 2 < q < p/(p - 1).
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1089–1097, August, 1998. 相似文献
15.
We obtain exact order estimates for trigonometric and orthoprojection widths of the Besov classes B
r
p,θ and Nikol’skii classes Hr p of periodic functions of many variables in the space L
q
for certain relations between the parameters p and q. 相似文献
16.
Pal-Andrej Nitsche 《Constructive Approximation》2006,24(1):49-70
We consider best N term approximation using anisotropic tensor product wavelet bases ("sparse grids"). We introduce a tensor
product structure ⊗q on certain quasi-Banach spaces. We prove that the approximation
spaces Aαq(L2) and Aαq(H1) equal tensor products of Besov spaces Bαq(Lq), e.g.,
Aαq(L2([0,1]d)) = Bαq(Lq([0,1])) ⊗q · ⊗q Bαq · ·(Lq([0,1])). Solutions to elliptic partial differential equations on polygonal/polyhedral domains belong to these new scales
of Besov spaces. 相似文献
17.
K. L. Avetisyan 《Potential Analysis》2008,29(1):49-63
We study anisotropic mixed norm spaces h(p,q,α) consisting of n-harmonic functions on the unit polydisc of by means of fractional integro-differentiation including small 0 < p < 1 and multi-indices α = (α
1,...,α
n
) with non-positive α
j
≤ 0. As an application, two different Bloch spaces of n-harmonic functions are characterized.
相似文献
18.
In this paper we give some criteria for the existence of compactly supported C
k+α-solutions (k is an integer and 0 ⩽ α < 1) of matrix refinement equations. Several examples are presented to illustrate the general theory. 相似文献
19.
To compute long term integrations for the pantograph differential equation with proportional delay qt, 0 < q ⩽ 1: y′(t) = ay(t) + by(qt) + f(t), y(0) = y
0, we offer two kinds of numerical methods using special mesh distributions, that is, a rational approximant with ‘quasi-uniform
meshes’ (see E. Ishiwata and Y. Muroya [Appl. Math. Comput., 2007, 187: 741-747]) and a Gauss collocation method with ‘quasi-constrained
meshes’. If we apply these meshes to rational approximant and Gauss collocation method, respectively, then we obtain useful
numerical methods of order p
* = 2m for computing long term integrations. Numerical investigations for these methods are also presented.
相似文献
20.
Let Ω ⊆ ℝn be a bounded convex domain with C
2 boundary. For 0 < p, q ⩽ ∞ and a normal weight φ, the mixed norm space H
k
p,q,φ
(Ω) consists of all polyharmonic functions f of order k for which the mixed norm ∥ · ∥p,q,φ < ∞. In this paper, we prove that the Gleason’s problem (Ω, a, H
k
p,q,φ
) is always solvable for any reference point a ∈ Ω. Also, the Gleason’s problem for the polyharmonic φ-Bloch (little φ-Bloch) space is solvable. The parallel results for the hyperbolic harmonic mixed norm space are obtained. 相似文献