On pointwise and analytic similarity of matrices

LetA(ε) andB(ε) be complex valued matrices analytic in ε at the origin.A(ε)≈ _p B(ε) ifA(ε) is similar toB(ε) for any |ε|<r,A(ε)≈_a B(ε) ifB(ε)=T(ε)A(ε)T ⁻¹(ε) andT(ε) is analytic and |T(ε)|≠0 for |ε|<r! In this paper we find a necessary and sufficient conditions onA(ε) andB(ε) such thatA(ε)≈ _a B(ε) provided thatA(ε)≈ _p B(ε). This problem arises in study of certain ordinary differential equations singular with respect to a parameter ε in the origin and was first stated by Wasow. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024     

Interpolating sequences in the ball ofC n

LetB be the unit ball ofC ⁿ , I give necessary conditions on sequenceS of points inB to beH ^∞(B) interpolating in term of aC ⁿ valued holomorphic function zero onS (a substitute for the interpolating Blaschke product). These conditions are sufficient to prove that the sequenceS is interpolating for ∩ _p>1 (B) and is also interpolating forH ^p (B) for 1≤p<∞.     

Extremal probabilities for Gaussian quadratic forms

 Denote by Q an arbitrary positive semidefinite quadratic form in centered Gaussian random variables such that E(Q)=1. We prove that for an arbitrary x>0, inf _Q P(Q≤x)=P(χ² _n /n≤x), where χ _n ² is a chi-square distributed rv with n=n(x) degrees of freedom, n(x) is a non-increasing function of x, n=1 iff x>x(1)=1.5364…, n=2 iff x[x(2),x(1)], where x(2)=1.2989…, etc., n(x)≤rank(Q). A similar statement is not true for the supremum: if 1<x<2 and Z ₁ ,Z ₂ are independent standard Gaussian rv's, then sup_0≤λ≤1/2 P{λZ ₁ ² +(1−λ)Z ₂ ² ≤x} is taken not at λ=0 or at λ=1/2 but at 0<λ=λ(x)<1/2, where λ(x) is a continuous, increasing function from λ(1)=0 to λ(2)=1/2, e.g. λ(1.5)=.15…. Applications of our theorems include asymptotic quantiles of U and V-statistics, signal detection, and stochastic orderings of integrals of squared Gaussian processes. Received: 24 June 2002 / Revised version: 26 January 2003 Published online: 15 April 2003 Research supported by NSA Grant MDA904-02-1-0091 Mathematics Subject Classification (2000): Primary 60E15, 60G15; Secondary 62G10     

Growth and Hölder conditions for the sample paths of Feller processes

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C _c ^∞(ℝ ⁿ )⊂D(A) and A|C _c ^∞(ℝ ⁿ ) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|_∞≤c(1+|ξ|²) and |Imp(x,ξ)|≤c ₀Rep(x,ξ). We show that the associated Feller process {X _t } _{t ≥0} on ℝ ⁿ is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β_∞ ^x :={λ>0:lim _{|ξ|→∞ | x − y |≤2/|ξ|}|p(y,ξ)|/|ξ|^λ=0} or δ_∞ ^x :={λ>0:liminf _{|ξ|→∞ | x − y |≤2/|ξ| |ε|≤1}|p(y,|ξ|ε)|/|ξ|^λ=0}, and obtain a.s. (ℙ ^x ) that lim _{t →0} t ^−1/λ _{s ≤ t} |X _s −x|=0 or ∞ according to λ>β_∞ ^x or λ<δ_∞ ^x . Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27]. Received: 21 July 1997 / Revised version: 26 January 1998     

The Besov and Triebel-Lizorkin spaces with high order on the Lipschitz curves

The Besov spacesB _p ^α,μ (Γ) and Triebel-Lizorkin spacesF _p ^α,μ (Γ) with high order α∈R on a Lipschitz curve Γ are defined. when 1≤p≤∞, 1≤q≤∞. To compare to the classical case a difference characterization of such spaces in the case |α|<1 is given also. The author is supported in part by the Foundation of Zhongshan University Advanced Research Centre and NSF of China.     

Some consequences of Pisier’s approach to interpolation

A recent result of Pisier onK-functionals is exploited in the context of function spaces defined from singular integrals. The main consequences appear forH ^p spaces on the complex ball (1≤p≤∞) and spaces of differentiable functions in several variables. It turns out that the complex variable arguments used in [P1] may in certain cases be substituted for real variable methods, which of course have a wider range of applicability. We discuss interpolation of vector valued Hardy spaces on the ball and prove thatL ¹(S)/H ¹(B) satisfies Grothendieck’s theorem, similarly as in the disc case.     

Many partition relations below density

We force 2 ^λ to be large, and for many pairs in the interval (λ, 2 ^λ ) a strong version of the polarized partition relations holds. We apply this to problems in general topology. For example, consistently, every 2 ^λ is the successor of a singular and for every Hausdorff regular space X, hd(X) ≤ s(X)⁺³, hL(X) ≤ s(X)⁺³ and better when s(X) is regular, via a halfgraph partition relations. For the case s(X) = ℵ ₀ we get hd(X), hL(X) ≤ N ₂.     

Isolation and Component Structure in Spaces of Composition Operators

We establish a condition that guarantees isolation in the space of composition operators acting between H^p(B_N) and H^q(B_N), for 0 < p ≤ ∞, 0 < q < ∞, and N ≥ 1. This result will allow us, in certain cases where 0 < q < p ≤ ∞, completely to characterize the component structure of this space of operators.     

Completeness of the system {z^λn} in L^2[B,α0]

Let B be an unbounded domain located outside an angle domain with vertex at the origin, A ={λn}（n = 1,2,...） be a sequence of complex numbers satisfying sup ｜ arg（λn）｜ 〈 α 〈 π/2 and denote by M（∧） = {z^λ, λ ∈ ∧} the corresponding system of functions z^λ（λ∈∧）. Let α0（z） be a weight function defined on B. We obtain a completeness theorem for the system M（∧） in the Hilbert space L^2 [B, α0].     

The convergence of multidimensional fourier-bessel series

We study the convergence of series of eigenfunctions of the Laplacian in the unit ballB ^d. The problem is posed in the spacesL _rad ^p (L _ang ² ). A convergence result is obtained in the sharp range2d/(d + 1) <p <2d/(d-1). There is a close connection with the spherical summation of classical trigonometric expansions. The proofs involve weighted inequalities for singular integrals, as well as a precise decomposition of oscillatory integrals using van der Corput’s method.     

On the Erlang Loss Model with Time Dependent Input

We consider the M(t)/M(t)/m/m queue, where the arrival rate λ(t) and service rate μ(t) are arbitrary (smooth) functions of time. Letting p_n(t) be the probability that n servers are occupied at time t (0≤ n≤ m, t > 0), we study this distribution asymptotically, for m→∞ with a comparably large arrival rate λ(t) = O(m) (with μ(t) = O(1)). We use singular perturbation techniques to solve the forward equation for p_n(t) asymptotically. Particular attention is paid to computing the mean number of occupied servers and the blocking probability p_m(t). The analysis involves several different space-time ranges, as well as different initial conditions (we assume that at t = 0 exactly n₀ servers are occupied, 0≤ n₀≤ m). Numerical studies back up the asymptotic analysis. AMS subject classification: 60K25,34E10 Supported in part by NSF grants DMS-99-71656 and DMS-02-02815     

The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup

Let (ℋ _t ) _t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix λ(R−I), where λ>0 and R is a skew-adjoint matrix, and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L ²(γ _∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ_* f(x)=sup  _t≥o |ℋ _t f(x)| is of weak type 1 with respect to the invariant measure γ _∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L ^p (γ _∞) if and only if 1<p≤∞.      

Analyticity of the sine family with L p -maximal regularity for second order Cauchy problems

If the second order problem u（t） ＋ Bu（t） ＋ Au（t） = f（t）, u（0） =u（0） = 0 has L^p-maximal regularity, 1 〈 p 〈 ∞, the analyticity of the corresponding propagator of the sine type is shown by obtaining the estimates of ‖λ（λ^2 ＋ λB ＋ A）^-1‖ and ‖B（λ^2 ＋ λB ＋ A）^-1‖ for λ∈ C with Reλ 〉 ω, where the constant ω≥ 0.     

Toeplitz Operators on Generalized Bergman Spaces

We consider the weighted Bergman spaces HL²(\mathbb B^d, m_l){\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}, where we set dm_l(z) = c_l(1-|z|²)^l dt(z){d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}, with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.     

Generating Uniform Random Vectors in Z p k : The General Case

We consider the rate of convergence of the Markov chain X _n+1=A X _n +B _n (mod p), where A is an integer matrix with nonzero eigenvalues, and {B _n } _n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Q ^k invariant under A. If for all eigenvalues λ _i of A, then n=O((ln p)²) steps are sufficient and n=O(ln p) steps are necessary to have X _n sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λ _i that are roots of positive integer numbers, |λ ₁|=1 and |λ _i |>1 for all , then O(p ²) steps are necessary and sufficient.      

Duals of holomorphic Besov spaces on the polydisks and diagonal mappings

Let U ⁿ be the unit polydisk in C ⁿ and S be the space of functions of regular variation. Let 1 ≤ p < ∞, ω = (ω ₁, ..., ω _n ), ω _j ∈ S(1 ≤ j ≤ n) and f ∈ H(U ⁿ ). The function f is said to be in holomorphic Besov space B _p (ω) if

Properties of generalised juxtapolynomials

GivenF(z),f ₁(z), ..,f _n(z) defined on a finite point setE, and givenB — the set of generalised polynomials Σ _k ^=1/n a _kf_k(z) — the definition of a juxtapolynomial is extended in the following manner: for a fixedλ(0<λ≦1),f(z) εB is called a generalizedλ-weak juxtapolynomial toF(z) onE if and only if there exists nog(z) εB for whichg(z)=F(z) wheneverf(z)=F(z) and |g(z)−F(z) |<λ|f(z)−F(z)| wheneverf(z)≠F(z). The properties of suchf(z) are investigated with particular attention given to the real case. This note is an extension of a part of the author’s M.Sc. Thesis under the supervision of Prof. B. Grünbaum to whom the author wishes to express his sincerest appreciation. The author also wishes to thank Dr. J. Lindenstrauss for his valuable remarks in the preparation of this paper.     

On simple modules of Hecke algebras of type Dn

This is a continuation of our previous work. We classify all the simple ℋ_q(D _n )-modules via an automorphismh defined on the set { λ | D^λ ≠ 0}. Whenf _n(q) ≠ 0, this yields a classification of all the simple ℋ ^q (D _n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ⁽¹⁾ = λ⁽²⁾,wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ℋ_q,1(B _n )- module D^λ remains irreducible on restriction to ℋ_q(D _n ).     

Discrete characterization of the Paley-Wiener space with several variables

In this paper, we obtain a characterization of the Paley-Wiener space with several variables, which is denoted byB _{π, p} (R ⁿ ), 1≤p<∞, i.e., for 1<p<∞,B _{π, p} (R ⁿ ) is isomorphic tol _p (Z ⁿ ), and forp=1,B _{π, 1} (R ⁿ ) is isomorphic to the discrete Hardy space with several variables, which is denoted byH(Z ⁿ ). This project is supported by the National Natural Science Foundation of China (19671012) and Doctoral Programme Institution of Higher Education Foundation of Chinese Educational Committee and supported by Youth Foundation of Sichuan.     

非对称Cantor集的Hausdorff中心测度

§ 1　 IntroductionThe class of Cantor sets is a typical one of sets in fractal geometry.Mathematicianshave paid their attentions to such sets for a long time.Itis well known that the Hausdorffmeasure of the Cantor middle- third set is1(see[1]) .Recently,Feng[3] obtained the exactvalues of the packing measure for a class of linear Cantor sets.Using Feng s method,Zhuand Zhou[5] obtained the exactvalue of Hausdorff centred measure of the symmetry Cantorsets.In this papar,we consider the Ha…