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1.
We prove that the singular numbers of the Cauchy transform
onL
2(D) are asymptotically
, whiles
n
(C
|
L
a
2
(D))1/n (whereL
a
2
(D) is the subspace of analytic functions inL
2(D)). Also, the singular numbers of the logarithmic potential
onL
2(D) are asympoticallys
n
(L)1/n, whiles
n(L
|L
a
2
(D))1/n
2. Our methods yield the asymptotic behavior of the singular numbers of the Cauchy Transform fromL
L
2
() intoL
2() where and are rotation-invariant measures on
.The author was partly supported by a grant from the national Science Foundation. 相似文献
2.
LetB be a separable Banach space and let {:||1} denote the unit ball ofB
*. LetX be a symmetricp-stableB-valued random variable and let {X
j
}
j=1
n
be i.i.d. copies ofX. LetB
1 be a finite-dimensional Banach space with a symmetric unconditional basis {y
j
}
j=1
n
. An upper bound is obtained for
that improves the one given by Giné, Marcus and Zinn [J. Functional Anal.
63, 47–73 (1985)]. 相似文献
3.
J. Hastad 《Combinatorica》1988,8(1):75-81
We prove that given a point
outside a given latticeL then there is a dual vector which gives a fairly good estimate for how far from the lattice the vector is. To be more precise, there is a set of translated hyperplanesH
i, such thatL
iHi andd(
iHi)(6n
2+1)–1
d(
,L).Supported by an IBM fellowship. 相似文献
4.
In this paper we study integral operators of the form
1 + ... + m = n. We obtain the L
p
(w) boundedness for them, and a weighted (1, 1) inequality for weights w in A
p
satisfying that there exists c 1 such that w(a
i
x) cw(x) for a.e. x n, 1 i m. Moreover, we prove
for a wide family of functions f L
(n).Partially supported by CONICET, Agencia Cordoba Ciencia and SECYT-UNC. 相似文献
5.
Given two disjoint subsets T
1 and
T
2 of
nodes in an undirected 3-connected graph G = (V, E) with node set
V and arc set
E, where
and
are even numbers, we
show that V can be
partitioned into two sets V
1 and
V
2
such that the graphs induced by V
1 and
V
2 are
both connected and
holds for each
j = 1,2. Such a partition can
be found in
time. Our proof relies
on geometric arguments. We define a new type of convex
embedding of k-connected
graphs into real space R
k-1 and prove that for
k = 3 such an embedding
always exists.
1 A preliminary version
of this paper with title Bisecting Two Subsets in 3-Connected
Graphs appeared in the Proceedings of the 10th Annual
International Symposium on Algorithms and Computation, ISAAC
99, (A. Aggarwal, C. P. Rangan, eds.), Springer LNCS 1741,
425–434, 1999. 相似文献
6.
Georgi R. Grozev 《Numerische Mathematik》1989,56(4):331-343
Summary LetLM
N be the set of allL-monosplines withN free knots, prescribed by a pair (x;E) of pointsx = {x
i
}
1
n
,a <x
1 < ... <x
n
<b and an incidence matrixE = (e
ij
)
i=1
n
,
r-1
j=0
with
Denote byLM
N
O
the subset ofLM
N consisting of theL-monosplines withN simple knots (n=N). We prove that theL-monosplines of minimalL
p-norms inLM
N belong toLM
N
O
.The results are reformulated as comparison theorems for quadrature formulae. 相似文献
7.
Summary We consider a (possibly) vector-valued function u: RN, Rn, minimizing the integral
, 2-2/(n*1)<p<2, whereD
i
u=u/x
i
or some more general functional retaining the same behaviour, we prove higher integrability for Du: D1 u,..., Dn–1 u Lp/(p-1) and Dnu L2; this result allows us to get existence of second weak derivatives: D(D1 u),...,D(Dn–1u)L2 and D(Dn u) L
p.This work has been supported by MURST and GNAFA-CNR. 相似文献
8.
Erwin Bolthausen 《Probability Theory and Related Fields》1978,43(2):173-181
Summary LetI
2 be the unit cube of
andX
i be independentI
2-valued random variables that are distributed according to Lebesgue-measure. IfS is the set of closed convex subsets ofI
2 we consider the process
n
(A)
AS,where
.It is proved that this process suitably normalized converges in a suitable weak sense to a Gaussian process. 相似文献
9.
K. Sikorski 《Numerische Mathematik》1984,43(2):225-240
Summary For a given nonnegative we seek a pointx
* such that |f(x
*)| wheref is a nonlinear transformation of the cubeB=[0,1]
m
into (or
p
,p>1) satisfying a Lipschitz condition with the constantK and having a zero inB.The information operator onf consists ofn values of arbitrary linear functionals which are computed adaptively. The pointx
* is constructed by means of an algorithm which is a mapping depending on the information operator. We find an optimal algorithm, i.e., algorithm with the smallest error, which usesn function evaluations computed adaptively. We also exhibit nearly optimal information operators, i.e., the linear functionals for which the error of an optimal algorithm that uses them is almost minimal. Nearly optimal information operators consists ofn nonadaptive function evaluations at equispaced pointsx
j
in the cubeB. This result exhibits the superiority of the T. Aird and J. Rice procedure ZSRCH (IMSL library [1]) over Sobol's approach [7] for solving nonlinear equations in our class of functions. We also prove that the simple search algorithm which yields a pointx
*=x
k
such that
is nearly optimal. The complexity, i.e., the minimal cost of solving our problem is roughly equal to (K/)
m
. 相似文献
10.
Hansmartin Zeuner 《Journal of Theoretical Probability》1994,7(2):225-245
Let (X
n:n) be i.i.d. with finite variance and values in a hypergroupK:=+ or and
j=1
n
X
j
be the randomized sum of these random variables. It is shown that the processes
converge in distribution to a Gaussian process in the caseK=+, that the processes
converge towards a Bessel process on + in the case of polynomial growth of the hypergroupK=+ or , and that in the case of exponential growth
converges towards a Brownian motion asn. 相似文献
11.
For a closed normal subgroupN of a locally compact groupG view a closed subset
of Prim*
L
1
(G/N) as a subsetE of Prim*
L
1
(G) in the canonical way and writeN
for Prim*
L
1
(G/N) as a subset of Prim*
L
1
(G); then the injection theorem says: IfE is spectral (i.e. of synthesis), then
is so; and if
andN
are spectral, thenE is too. In case of a group of polynomial growth with symmetricL
1-algebra, where smallest idealsj (E) with given hulls exist, it is known thatN
is always spectral. For a closed,G-invariant subsetF of Prim*
L
1
(N) define a closed subsetE of Prim*
L
1
(G) by
. Denote by e (I') the ideal generated byC
00
(G)*I', where theG-invariant idealI' ofL
1
(N) is viewed as a subset of measures onG, then the projection theorem states: IfE is spectral, thenF is so, and ifF is spectral withe (j (F))=j (E) thenE is spectral. All assumptions are fulfilled for instance, ifG andN are of polynomial growth with symmetricL
1-algebra and eitherSIN-groups or solvable. 相似文献
12.
Chin-Cheng Lin 《逼近论及其应用》2001,17(2):45-53
Let fL
p (R), 1pt8, and c
j
be the inner product of f and the Hermite function h
j
. Assume that c
j
's satisfy
If r=5/4, then the Hermite series c
j
h
j
conerges to f almost everywhere. If r=9/4-1/p, the c
j
h
j
converges to f in L
p (R). 相似文献
13.
N. A. Shirokov 《Journal of Mathematical Sciences》2005,129(4):4083-4086
Let
n be the unit ball in ℂn, n ≥ 2. Let Tα = {z ∈
n : (z, a) = |a|2} for a ∈
n and denote
for a discrete set A in
n. We find a sharp necessary condition for a set A to be a part of the zero-set for a function in H∞(
n). Bibliography 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 272–278. 相似文献
14.
Joseph Rosenblatt 《Mathematische Annalen》1977,230(3):245-272
For a mean zero norm one sequence (f
n
)L
2[0, 1], the sequence (f
n
{nx+y}) is an orthonormal sequence inL
2([0, 1]2); so if
, then
converges for a.e. (x, y)[0, 1]2 and has a maximal function inL
2([0, 1]2). But for a mean zerofL
2[0, 1], it is harder to give necessary and sufficient conditions for theL
2-norm convergence or a.e. convergence of
. Ifc
n
0 and
, then this series will not converge inL
2-norm on a denseG
subset of the mean zero functions inL
2[0, 1]. Also, there are mean zerofL[0, 1] such that
never converges and there is a mean zero continuous functionf with
a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c
n
| = 0(n
–) for >1/2, then
converges a.e. and unconditionally inL
2[0, 1]. In addition, for any mean zerof of bounded variation, the series
has its maximal function in allL
p[0, 1] with 1p<. Finally, if (f
n
)L
[0, 1] is a uniformly bounded mean zero sequence, then
is a necessary and sufficient condition for
to converge for a.e.y and a.e. (x
n
)[0, 1]. Moreover, iffL
[0, 1] is mean zero and
, then for a.e. (x
n
)[0, 1],
converges for a.e.y and in allL
p
[0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one. 相似文献
15.
Elliott H. Lieb 《Inventiones Mathematicae》1983,74(3):441-448
IfA andB are two bounded domains in
n
and (A), (B) are the lowest eigenvalues of – with Dirichlet boundary conditions then there is some translate,B
x, ofB such that (AB
x)<(A)+(B). A similar inequality holds for
.There are two corollaries of this theorem: (i) A lower bound for sup
x
{volume (AB
x)} in terms of (A), whenB is a ball; (ii) A compactness lemma for certain sequences inW
1,p
(
n
).Work partially supported by U.S. National Science Foundation grant PHY-8116101 A01. AMS(MOS) Classification: 35P15 相似文献
16.
The aim of this paper is to prove Paley type inequalities for two-parameter Vilenkin system. Our main result is the following estimate:
for martingales f H
p
(G
p
× G
q
) (0 < p 1). Here G
p
and G
q
are Vilenkin groups generated by the sequences p = (p
n
) and q = (q
n
), respectively, and f^(u, v) (u, v N) is the (u,v)th (two-parameter) Vilenkin-Fourier coefficient of f. The Hardy space H
p
(G
p
× G
q
) is defined by means of a usual martingal maximal function.We get the inequality (*) from its dual version, especially it follows from a BMO-result in the case p = 1. Furthermore, interpolation leads to an L
p
-variant of (*) for 1 < p 2. We also formulate an analogous statement for another Hardy space. In the so-called unbounded case, i.e. when p or q is not bounded, we shall investigate whether (*) can be improved. Our results hold also in the case of higher dimensions. 相似文献
17.
In this paper we are concerned with the summability of the geometric series
by matrix methods. We prove the following theorem: Suppose Mo:={z:|z|<1}, M1, M2, is a collection of countably many Lebesgue measureable, disjoint sets. For k=1,2, let fk be a prescribed function, analytic on
. Then there exists a triangular matrix
, such that the V-transform {n(z)} of the geometric series has the following properties: {n(z)} converges compactly to
on Mo; for k=1,2, there are sets Bk, such that
has Lebesgue-measure zero and n(z)fk(z) for zBk; if
there is a set B*, such that B*M* has Lebesgue-measure zero and {n(z)} diverges for zB*. 相似文献
18.
Margarita Ramalho 《Algebra Universalis》1985,20(2):243-253
For a pseudocomplemented latticeL, we prove that the filter Dn(L), 1n<, generated by then-strongly dense elements is contained in everyn-normal filter. Hence, Dn(L)=Gn(L)=Radn (L), where Gn(L) is the intersection of all n-normal filters, and Radn (L) is the intersection of alln-normal prime filters. Moreover, we prove that a prime filterP is n-normal iff Dn(L)=P. Consequently, for
, we have Dn(L)=Gn(L)=Radn (L) and therefore
iff Radn(L)={1} (or iff Gn(L)={1}).Considering the skeleton S(L) ofL, a complete clarification of the relationship between filters ofL and S(L) is given by studying th correspondence FFS(L).We state that D(L) (and that D1(L), if
is an irredundant intersection of maximal filters (resp. of *-maximal filters) iff S(L) is finite.Finally, for
we state that the least *-congruence for which
is that one generated by Dn(L).Presented by B. Jónsson.Research supported by the I.N.I:C, (Centro de Algebra da Universidade de Lisboa). 相似文献
19.
S. T. Norvidas 《Journal of Mathematical Sciences》1990,49(4):1095-1100
The paper is devoted to the investigation of the norms
of the operators F(D), where
, in the Banach spaces Bk, consisting of the restrictions toR
n
of the entire functions inC
n
, whose Fourier transforms, understood in the sense of distribution theory, are concentrated on a compactum K R
n
. The class from whichone recruits the functions F is the class of the Fourier transforms of regular Borel measures with finite variation onR
n
.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 49, pp. 86–93, 1988. 相似文献