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1.
If w1,…,w
N is a finite sequence of nonzero points in the unit disk, then there are distinct points λ1,…, λN on the unit circle and positive numbers Μ1,…,Μ
N such that
is the zero sequence of the function 1 —
. The points λ1,…, λN and numbers Μ1,…,ΜN are unique (except for reorderings). 相似文献
2.
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
3.
A powerful tool for studying the growth of analytic and harmonic functions is Hall's Lemma, which states that there is a constantC>0 so that the harmonic measure of a subsetE of the closed unit disk
evaluated at 0 satisfies
whereE
rad is the radial projection ofE onto
. FitzGerald, Rodin and Warschawski proved that ifE is a continuum in
whose radial projection has length at most π then (*) is true withC=1, and they asked how large the length, |E
rad|, can be in order for their result to be valid. We prove that (*) holds withC=1 for every continuum
satisfying
and θc cannot be replaced by a larger number. Fuchs asked for the largest constantC so that (*) holds for allE. We show that for every continuum
, (*) holds withC=C
2π≅.977126698498665669…, whereC
2π is the harmonic measure of the two long sides of a 3∶1 rectangle evaluated at the center. There are Jordan curves for which
equality holds in (*) withC=C
2π.
The authors are supported in part by NSF grants DMS-9302823 and DMS-9401027, and while at MSRI by NSF grant DMS-9022140. 相似文献
4.
T. Levasseur 《Transformation Groups》1998,3(4):337-353
Let
u
be a compact Lie algebra and let
u
be its complexification. Let ζ−1/2 be the inverse on the set of regular elements of
u
of a square root of the discriminant of
. Generalizing a result of W. Lichtenstein in the case
u
=
(n, ℂ) or
(nℝ), we prove that ∂(q).ζ1/2 is non zero for all harmonic polynomialsq ∈S(
) \ {0}. This fact is deduced from results about equivariantD-modules supported on the nilpotent cone of
. 相似文献
5.
By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation
given by ( a1, …, an). Assume that either a1, …, an are positive rational numbers or
for some
Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ℝn must be infinite 相似文献
6.
Peter Raith 《Journal d'Analyse Mathématique》1999,78(1):117-142
Assume thatX is a finite union of closed intervals and consider aC
1-mapX→ℝ for which {c∈X: T′c=0} is finite. Set
. Fix ann ∈ ℕ. For ε>0, theC
1-map
is called an ε-perturbation ofT if
is a piecewise monotonic map with at mostn intervals of monotonicity and
is ε-close toT in theC
1-topology. The influence of small perturbations ofT on the dynamical system (R(T),T) is investigated. Under a certain condition on the continuous functionf:X → ℝ, the topological pressure is lower semi-continuous. Furthermore, the topological pressure is upper semi-continuous for
every continuous functionf:X → ℝ. If (R(T),T) has positive topological entropy and a unique measure μ of maximal entropy, then every sufficiently small perturbation
ofT has a unique measure
of maximal entropy, and the map
is continuous atT in the weak star-topology. 相似文献
7.
Anne Cumenge 《Arkiv f?r Matematik》2001,39(1):1-25
Let Ω be a bounded convex domain in C
n
, with smooth boundary of finite typem.
The equation
is solved in Ω with sharp estimates: iff has bounded coefficients, the coefficients of our solutionu are in the Lipschitz space Λ. Optimal estimates are also given when data have coefficients belonging toL
p(Ω),p≥1.
We solve the
-equation by means of integral operators whose kernels are not based on the choice of a “good” support function. Weighted
kernels are used; in order to reflect the geometry ofbΩ, we introduce a weight expressed in terms of the Bergman kernel of Ω. 相似文献
8.
LetW be an open Riemann surface and
ap-sheeted (1<p<∞) unlimited covering surface ofW. Denote by Δ1 (resp.,
) the minimal Martin boundary ofW (resp.,
). For ζ ∈ Δ, let
ζ be the (cardinal) number of the set of pionts
which lie over ζ and
the class of open connected subsetsM ofW such thatM∪{ζ} is a minimal fine neighborhood of ζ. Our main result is the following:
, where
is the number of components of π-1
M and π is the projection of
ontoW. Moreover, some applications of the above results are discussed whenW is the unit disc. 相似文献
9.
Moshe Dubiner 《Journal d'Analyse Mathématique》1995,67(1):39-116
We consider the problem of polynomial approximation to a real valued functionf defined on a compact set
. An approximation theorem is proven in terms of the newly defined modulus of approximation. It is shown to imply a multidimensional
Jackson type theorem which is stronger than previously known results even for the interval [−1, 1]. A strong multidimensional
Bernstein type inverse theorem is also proven. We allow quite general approximation quasi-norms including
for 0<q≤∞.
We have found that the space of polynomials ℙ on a compact setX induces a semimetric
which encapsulates the local structure of ℙ. Any semimetric ρ equivalent to
suffices for the rough theory presented here. Many examples of sets
and their metrics are presented. 相似文献
10.
We show that for a
-action Ψ being the Kronecker sum of a symbolic strictly ergodic
-actionT and a Chacon
-actionS, the rank (covering number) of Ψ is the same as that forT. Using this result we construct, for a given natural numberr≥2 and a real numberb∈(0,1) withr\b≥1, a
-action with rankr, covering numberb and a simple spectrum. On the other hand, for any positive integersr, m with 1≤m≤r≤∞ we construct a
-action with rankr and spectral multiplicitym. 相似文献
11.
V. V. Andrievskii 《Journal d'Analyse Mathématique》2005,96(1):283-295
LetG⊂C be a quasidisk,K ⊂ G be a compact set, andp
n be a non-constant complex polynomial of degree at mostn. We establish the inequality
whereα
n < 0 depends onn, K,
and the geometrical structure of ϖG. 相似文献
12.
Ivan Č. Jovanović Eberhard Malkowsky Vladimir Rakočević 《Rendiconti del Circolo Matematico di Palermo》2003,52(2):308-314
LetF be aBK space withAK and
denote the set of all formal power series
with
such that
ε F for the sequence of coefficients of
. We give a necessary and sufficient condition for a point to be a bounded point evaluation on
, and for a polynomial to be cyclic in
. As special cases, we obtain the results for the space ℓ
p
(β) in [7].
Research of the authors supported under the research project #1232 of the Serbian Ministry of Sciences and Tecnology and,
in the case of the second author, also by the DAAD foundation (German Academic Exchange Service), grant 911 103 102 8. 相似文献
13.
Hoai-Minh Nguyen 《Journal d'Analyse Mathématique》2007,101(1):367-395
In this paper, we prove the estimate
, for every δ ∈ (0, ℓN), where C = C(N) is a positive constant depending only on N and
. We show that the constant ℓN in this estimate is optimal. We also present a class of maps from
into
, strictly larger than
, on which we can define the notion of degree and for which the previous inequality still holds. 相似文献
14.
Fabio Vlacci 《Journal d'Analyse Mathématique》1998,75(1):51-66
We introduce the notion of pointwise regularity (
) of Colombeau’s generalized functions and give comparison theorems between
regularity,
- andC
∞-regularity. We also define the notion of a pointwise wave front set and establish a theorem concerning the effect of a linear
generalized partial differential operator on such a wave front. 相似文献
15.
Summary Let
be a sequence of independent identically distributed random variables withθ
1∼G and the conditional distribution ofx
1 givenθ
1=θ given by
. HereG is unknown andF
θ(·) is known. This paper provides estimators
ofG based onx
1, …,x
n such that the random variable sup
has an asymptotic distribution asn→∞ under certain on conditionsG and for certain choices ofF
θ. A simulation model has been discussed involving the uniform distribution on (0, θ) forF
θ and an exponential distribution forG.
Research supported by the National Science Foundation under Grant #MCS77-26809. 相似文献
16.
We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of
in the limitN→∞. The resulting Hopf algebra Rep
is a tensor product of its Hopf subalgebras Repa
,a ∈ ℂ×/q2ℤ. Whenq is generic (resp.,q
2 is a primitive root of unity of orderl), we construct an isomorphism between the Hopf algebra Rep
a
and the algebra of regular functions on the prounipotent proalgebraic group
(resp.,
). Whenq is a root of unity, this isomorphism identifies the Hopf subalgebra of Rep
a
spanned by the modules obtained by pullback with respect to the Frobenius homomorphism with the algebra generated by the
coefficients of the determinant of an element of
considered as anl×l matrix over the Taylor series. This gives us an explicit formula for the Frobenius pullbacks of the fundamental representations.
In addition, we construct a natural action of the Hall algebra associated to the infinite linear quiver (resp., the cyclic
quiver withl vertices) on Rep
a
and describe the span of tensor products of evaluation representations taken at fixed points as a module over this Hall algebra. 相似文献
17.
We study polymodal logics with n modal connectives □1,...,□n, each of which satisfies the axioms of S5 and, moreover, obeys the commutativity laws
. The following results are proved: (1) the logic S5nC is not locally finite; (2) the inference rule A(p1, …, pm)/B(p1, …, pm) is not admissible in
, and on a one-element model ∉, there exists a valuation of variables p1, …, pm, such that ∉ ⊪ A.
Supported by RFFR grant No. 96-01-00228.
Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 483–493, September–October, 1997. 相似文献
18.
Bounds for the 3G-expression∫
Ω
G(x,z)G(z,y)d,z/G(x,y) play a fundamental role in potential theory. Here,G(x,y) is the Green function for the Laplace problem with zero dirichlet boundary conditions on Ω. The 3G-formula equals
, the expected lifetime for a Brownian motion starting in
that is killed on exiting ω and conditioned to converge to and to be stopped at
. Although it was shown by probabilistic methods for bounded (simply connected) 2d-domains that ifx ε δΩ, then the supremum ofy \at E
x
y
is assumed for somey at the boundary, the analogous question remained open forx in the interior. Here we are able to give an answer in the case thatB ⊂ ℝ is the unit disk. The dependence of this quantity on the positions ofx andy is investigated, and it is shown that indeed E
x
y
(\gt\om) is maximized on
by opposite boundary points. The result also gives an answer to a number of questions related to the best constant for the
positivity-preserving property of some elliptic systems. In particular, it confirms a, relationE
x
y
(\gt\om) with a ‘sum of inverse eigenvalues’ that was conjectured recently by Kawohl and Sweers. 相似文献
19.
We consider two-phase metrics of the form ϕ(x, ξ) ≔
, where α,β are fixed positive constants and B
α, B
β are disjoint Borel sets whose union is ℝN, and prove that they are dense in the class of symmetric Finsler metrics ϕ satisfying
. Then we study the closure
of the class
of two-phase periodic metrics with prescribed volume fraction θ of the phase α. We give upper and lower bounds for the class
and localize the problem, generalizing the bounds to the non-periodic setting. Finally, we apply our results to study the
closure, in terms of Γ-convergence, of two-phase gradient-constraints in composites of the type f(x, ∇ u) ≤ C(x), with C(x) ∈ {α, β } for almost every x. 相似文献
20.
Yehoram Gordon 《Israel Journal of Mathematics》1981,39(1-2):141-144
LetX andY be Banach spaces. TFAE (1)X andY do not contain subspaces uniformly isomorphic to
(2) The local unconditional structure constant of the space of bounded operatorsL (X*k,Y
k) tends to infinity for every increasing sequence
and
of finite-dimensional subspaces ofX andY respectively. 相似文献