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1.
LetH
∞ be the algebra of bounded analytic functions in the unit diskD. LetI=I(f
1,...,f
N) be the ideal generated byf
1,...,f
N∈H
∞ andJ=J(f
1,...,f
N) the ideal of the functionsf∈H
∞ for which there exists a constantC=C(f) such that |f(z)|≤C(|f
1
(z)|+...;+|f
N
(z)|),z∈D. It is clear that
, but an example due to J. Bourgain shows thatJ is not, in general, in the norm closure ofI. Our first result asserts thatJ is included in the norm closure ofI ifI contains a Carleson-Newman Blaschke product, or equivalently, if there existss>0 such that
Our second result says that there is no analogue of Bourgain's example in any Hardy spaceH
p, 1≤p<∞. More concretely, ifg∈H
p and the nontangential maximal function of
belongs toL
p (T), theng is in theH
p-closure of the idealI.
Both authors are supported in part by DGICYT grant PB98-0872 and CIRIT grant 1998SRG00052. 相似文献
2.
LetA be the class of normalized analytic functions in the unit disk Δ and define the class
For a functionf εA the Alexander transformF
0 is given by
Our main object is to establish a sharp relation betweenβ andγ such thatf εP
β implies thatF
0 is starlike of orderγ, 0 ≤γ ≤ 1/2. A corresponding result for the Libera transformF
1(z) = 2∫
0
1
f(tz)dt is also given. 相似文献
3.
In this paper, the sharp estimates of all homogeneous expansions for f are established, where f(z) = (f
1(z), f
2(z), …, f
n
(z))′ is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in ℂ
n
and
$
\begin{gathered}
\frac{{D^{tk + 1} + f_p \left( 0 \right)\left( {z^{tk + 1} } \right)}}
{{\left( {tk + 1} \right)!}} = \sum\limits_{l_1 ,l_2 ,...,l_{tk + 1} = 1}^n {\left| {apl_1 l_2 ...l_{tk + 1} } \right|e^{i\tfrac{{\theta pl_1 + \theta pl_2 + ... + \theta pl_{tk + 1} }}
{{tk + 1}}} zl_1 zl_2 ...zl_{tk + 1} ,} \hfill \\
p = 1,2,...,n. \hfill \\
\end{gathered}
$
\begin{gathered}
\frac{{D^{tk + 1} + f_p \left( 0 \right)\left( {z^{tk + 1} } \right)}}
{{\left( {tk + 1} \right)!}} = \sum\limits_{l_1 ,l_2 ,...,l_{tk + 1} = 1}^n {\left| {apl_1 l_2 ...l_{tk + 1} } \right|e^{i\tfrac{{\theta pl_1 + \theta pl_2 + ... + \theta pl_{tk + 1} }}
{{tk + 1}}} zl_1 zl_2 ...zl_{tk + 1} ,} \hfill \\
p = 1,2,...,n. \hfill \\
\end{gathered}
相似文献
4.
E. Amar 《Journal of Geometric Analysis》1991,1(4):291-305
We show that if f1, f2 are bounded holomorphic functions in the unit ball
of ℂn such that
, |f1(z)|2 + |f2(z)2|2 ≥ δ2 >; 0, then any functionh in the Hardy space
,p < +∞ can be decomposed ash = f1h1
+ f2h2 with
. The Corona theorem in
would be the same result withp = +∞ and this question is still open forn ≳-2, but the preceding result goes in this direction. 相似文献
5.
For the problemP(λ): Maximizec
T
z subject toz∈Z(λ), whereZ(λ) is defined by an in general infinite set of linear inequalities, it is shown that the value-function has directional derivatives
at every point
such thatP(
) and its dual are both superconsistent. To compute these directional derivatives a min-max-formula, well-known in convex
programming, is derived. In addition, it is shown that derivatives can be obtained more easily by a limit-process using only
convergent selections of solutions ofP(λ
n
), λ
n
→
and their duals. 相似文献
6.
LetH be any complex inner product space with inner product <·,·>. We say thatf: ℂ→ℂ is Hermitian positive definite onH if the matrix
7.
Letf(z, t) be a subordination chain fort ∈ [0, α], α>0, on the Euclidean unit ballB inC
n. Assume thatf(z) =f(z, 0) is quasiconformal. In this paper, we give a sufficient condition forf to be extendible to a quasiconformal homeomorphism on a neighbourhood of
. We also show that, under this condition,f can be extended to a quasiconformal homeomorphism of
onto itself and give some applications.
Partially supported by Grant-in-Aid for Scientific Research (C) no. 14540195 from Japan Society for the Promotion of Science,
2004. 相似文献
8.
S. Treil 《Journal d'Analyse Mathématique》2002,87(1):481-495
The main result of the paper is that there exist functionsf
1,f
2,f inH
∞
satisfying the “corona condition”
9.
On Homogeneous Differential Polynomials
of Meromorphic Functions 总被引:2,自引:0,他引:2
In this paper, we study one conjecture proposed by W. Bergweiler and show that any
transcendental meromorphic functions f(z) have the form exp(αz+β) if f(z)f″(z)–a(f′ (z))2≠0,
where
. Moreover, an analogous normality criterion is obtained.
Supported by National Natural Science Foundation and Science Technology Promotion Foundation of Fujian
Province (2003) 相似文献
10.
In this paper we obtain a general lower bound for the tail distribution of the Fourier spectrum of Boolean functionsf on {1, −1}
N
. Roughly speaking, fixingk∈ℤ+ and assuming thatf is not essentially determined by a bounded number (depending onk) of variables, we have that
. The example of the majority function shows that this result is basically optimal. 相似文献
11.
Keiji Izuchi 《Journal d'Analyse Mathématique》1998,75(1):135-154
Letb be a Blaschke product with zeros {z
n
} in the open unit disk Δ. Let
be the set of sequences of non-negative integersp=(p
1,p
2,…) such that ∑
n=1
∞
p
n
(1 − |z
n
|) < ∞ andp
n
→∞ asn→∞. We study the class of weak infinite powers ofb,
Properties of these classes depend on the setS(b) of the cluster points in ∂Δ of {z
n
}. It is proved thatS(b)=∂Δ if and only if
, the Douglas algebra generated by
. Also, it is proved thatdθ(S(b))=0 if and only if there exists an interpolating Blaschke productB such that
. 相似文献
12.
Lou Yuanren 《分析论及其应用》1990,6(1):46-64
Let f∈Ap. For any positive integer l, the quantity Δ1,n−1(f:z) has been studied extensively. Here we give some quantitative estimates for
and investigate some pointwise estimates of Δ
l,n−1
(r)
(f;z).
Supported by National Science Foundation of China 相似文献
13.
In this paper we consider
, in one case that fx
0 (t) is a ΛBMV function on [0, ∞], and in another case thatfεL
1
m-1(Rn) and
when |k|=m−1 and f(x)=0 when |x−x0|<δ for some δ>0. Our theormes improve the results of Pan Wenjie ([1]). 相似文献
14.
The generalized Roper-Suffridge extension operator Ф(f) on the bounded complete Reinhardt domain Ω in Cn with n ≥ 2 is defined by Φrn,β2,γ2,…,βn,γn(f)(z)=(rf(z1/r),(rf(z1/r)/z1)β2(f'(z1/r))γ2z2,…,(rf(z1/r)/z1)βn(f'(z1/r)γnzn) for (z1,z2,…,zn) ∈Ω, where r = r(Ω) = sup{|z1| (z1,z2,…,zn) ∈ Ω},0 ≤ γj ≤ 1 -βj,0 ≤ βj ≤ 1,and we choose the branch of the power functions such that (f(z1)/z1)βj |z1=0 = 1 and (f′(z1))γj |z1=0 =1,j = 2,…,n. In this paper, we prove that the operator Фrn,β2,γ2,…,βn,γn(f) is from the subset of S*α(U) to S*α(Ω)(0 ≤ α < 1) on Ω and the operator Фrn,β2,γ2,…, βn,γn(f) preserves the starlikeness of order α or the spirallikeness of type β on Dp for some suitable constantsβj,γj,pj, where Dp ={(z1,z2,…,zn) ∈ Cn ∑nj=1|zj|pj < 1} (pj > 0, j = 1,2,…,n), U is the unit disc in the complex plane C, and Sα* (Ω) is the class of all normalized starlike mappings of order α on Ω. We also obtain that Φrn,β2,γ2,…,γn(f) ∈ S*α(Dp) if and only if f ∈ S*a(U) for 0 ≤ α < 1 and some suitable constants βj,γj,pj. 相似文献
15.
We give the “boundary version” of the Boggess-PolkingCR extension theorem. LetM andN be real generic submanifolds of ℂ
n
withN ⊂M and letV be a “wedge” inM with “edge”N and “profile” Σ ⊂T
NM in a neighborhood of a pointz
o.We identify in natural manner
and assume that for a holomorphic vector fieldL tangent toM and verifying
we have that the Levi form
takes a value
. Then we prove thatCR functions onV extend ∀ω to a wedgeV
1 “attached” toV in direction of a vector fieldiV such that |pr(iV(z
0))−iv
0| < ε (where pr is the projection pr:T
NX →T
MX |
N
).We then prove that when the Levi cone “relative to Σ”iZ
Σ = convex hull
is open inT
MX, thenCR functions extend to a “full” wedge with edgeN (that is, with a profile which is an open cone ofT
NX). Finally, we prove that iff is defined in a couple of wedges ±V with profiles ±Σ such thatiZ
Σ =T
MX, and is continuous up toN, thenf is in fact holomorphic atz
o. 相似文献
16.
Let Ω be a bounded smooth domain inR
2. Letf:R→R be a smooth non-linearity behaving like exp{s
2} ass→∞. LetF denote the primitive off. Consider the functionalJ:H
0
1
(Ω)→R given by
17.
A. A. Ryabinin 《Mathematical Notes》1998,64(5):629-633
The system
, where Λ={λ
n
} is the set of zeros (of multiplicitiesm
n
) of the Fourier transform
18.
Ki-Ho Kwon 《Israel Journal of Mathematics》1994,86(1-3):409-427
Suppose thatα>1, 0<R<∞ and thatf is analytic in |z|≤αR with |f(0)|≥1. It is shown that for a constant dα depending only onα,
. Therefore iff is entire of order λ<∞, logM(r,f)/T(r,f) has order at most λ/2. These results are shown by example to be quite precise. 相似文献
19.
In order to give an elementwise characterization of a subintegral extension of ℚ-algebras, a family of generic ℚ-algebras
was introduced in [3]. This family is parametrized by two integral parameters p ⩾ 0,N ⩾ 1, the member corresponding top, N being the subalgebraR = ℚ [{γn|n ⩾ N}] of the polynomial algebra ℚ[x1,…,x
p, z] inp + 1 variables, where
. This is graded by weight (z) = 1, weight (x
i) =i, and it is shown in [2] to be finitely generated. So these algebras provide examples of geometric objects. In this paper
we study the structure of these algebras. It is shown first that the ideal of relations among all the γn’s is generated by quadratic relations. This is used to determine an explicit monomial basis for each homogeneous component
ofR, thereby obtaining an expression for the Poincaré series ofR. It is then proved thatR has Krull dimension p+1 and embedding dimensionN + 2p, and that in a presentation ofR as a graded quotient of the polynomial algebra inN + 2p variables the ideal of relations is generated minimally by
elements. Such a minimal presentation is found explicitly. As corollaries, it is shown thatR is always Cohen-Macaulay and that it is Gorenstein if and only if it is a complete intersection if and only ifN + p ⩽ 2. It is also shown thatR is Hilbertian in the sense that for everyn ⩾ 0 the value of its Hilbert function atn coincides with the value of the Hilbert polynomial corresponding to the congruence class ofn. 相似文献
20.
K. F. Cheng 《Annals of the Institute of Statistical Mathematics》1982,34(1):479-489
Summary Letf
n
(p)
be a recursive kernel estimate off
(p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of
and show that the rate of almost sure convergence of
to zero isO(n
−α), α<(r−p)/(2r+1), iff
(r),r>p≧0, is a continuousL
2(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of
to zero under different conditions onf.
This work was supported in part by the Research Foundation of SUNY. 相似文献
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