Bergman-Type Reproducing Kernels, Contractive Divisors, and Dilations

Let Ω be a region in the complex plane. In this paper we introduce a class of sesquianalytic reproducing kernels on Ω that we call B-kernels. When Ω is the open unit disk and certain natural additional hypotheses are added we call such kernels k Bergman-type kernels. In this case the associated reproducing kernel Hilbert space (k) shares certain properties with the classical Bergman space L²_α of the unit disk. For example, the weighted Bergman kernels k^β_w(z)=(1−wz)^−β, 1β2 are Bergman-type kernels. Furthermore, for any Bergman-type kernel k one has H² (k)L²_a, where the inclusion maps are contractive, and M_ζ, the operator of multiplication with the identity function ζ, defines a contraction operator on (k). Our main results about Bergman-type kernels k are the following two: First, once properly normalized, the reproducing kernel for any nontrivial zero based invariant subspace of (k) is a Bergman-type kernel as well. For the weighted Bergman kernels k^β this result even holds for all _ζ-invariant subspace of index 1, i.e., whenever the dimension of /ζ is one. Second, if is any multiplier invariant subspace of (k), and if we set *= z , then M_ζ is unitarily equivalent to M_ζ acting on a space of *-valued analytic functions with an operator-valued reproducing kernel of the type

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where V is a contractive analytic function V :  → ( ,  *), for some auxiliary Hilbert space . Parts of these theorems hold in more generality. Corollaries include contractive divisor, wandering subspace, and dilation theorems for all Bergman-type reproducing kernel Hilbert spaces. When restricted to index one invariant subspaces of (k^β), 1β2, our approach yields new proofs of the contractive divisor property, the strong contractive divisor property, and the wandering subspace theorems and inner–outer factorization. Our proofs are based on the properties of reproducing kernels, and they do not involve the use of biharmonic Green functions as had some of the earlier proofs.     

Estimation de Yule–Walker d'un CAR(p) observé à temps discret

Let (X(lδ), l=0,n) be a discrete observation at mesh δ>0 of X, a CAR(p). Classical Yule–Walker estimation are biased and must be corrected. Resultant estimators converge if T=nδ→+∞, are asymptotically normal with rate , and efficient. The diffusion coefficient is also estimated, with rate .     

A functional Hungarian construction for sums of independent random variables

We develop a Hungarian construction for the partial sum process of independent non-identically distributed random variables. The process is indexed by functions f from a class , but the supremum over is taken outside the probability. This form is a prerequisite for the Komlós–Major–Tusnády inequality in the space of bounded functionals , but contrary to the latter it essentially preserves the classical n^−1/2logn approximation rate over large functional classes such as the Hölder ball of smoothness 1/2. This specific form of a strong approximation is useful for proving asymptotic equivalence of statistical experiments.     

The supremum of Brownian local times on Hölder curves

For , we consider L^f_t, the local time of space-time Brownian motion on the curve f. Let be the class of all functions whose Hölder norm of order α is less than or equal to 1. We show that the supremum of L^f₁ over f in is finite if α>1/2 and infinite if α<1/2.     

Probabilistic and average widths of multivariate Sobolev spaces with mixed derivative equipped with the Gaussian measure

We present sharp bounds on the Kolmogorov probabilistic (N,δ)-width and p-average N-width of multivariate Sobolev space with mixed derivative

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, equipped with a Gaussian measure μ in , that is where 1<q<∞,0<p<∞, and ρ>1 is depending only on the eigenvalues of the correlation operator of the measure μ (see (4)).     

Nonradial large solutions of sublinear elliptic problems

Let p be a nonnegative locally bounded function on , N3, and 0<γ<1. Assuming that the oscillation sup_|x|=rp(x)−inf_|x|=rp(x) tends to zero as r→∞ at a specified rate, it is shown that the equation Δu=p(x)u^γ admits a positive solution in satisfying lim_|x|→∞u(x)=∞ if and only if

     

Pointwise universal trigonometric series

A series is called a pointwise universal trigonometric series if for any , there exists a strictly increasing sequence of positive integers such that converges to f(z) pointwise on . We find growth conditions on coefficients allowing and forbidding the existence of a pointwise universal trigonometric series. For instance, if as |n|→∞ for some ε>0, then the series S_a cannot be pointwise universal. On the other hand, there exists a pointwise universal trigonometric series S_a with as |n|→∞.     

Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel–Leader graphs

We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel–Leader graph , where q,r2. The latter is the horocyclic product of two homogeneous trees with respective degrees q+1 and r+1. When q=r, it is the Cayley graph of the wreath product (lamplighter group) with respect to a natural set of generators. We describe the full Martin compactification of these random walks on -graphs and, in particular, lamplighter groups. This completes previous results of Woess, who has determined all minimal positive harmonic functions.     

The Diameter of Sparse Random Graphs

We consider the diameter of a random graph G(n, p) for various ranges of p close to the phase transition point for connectivity. For a disconnected graph G, we use the convention that the diameter of G is the maximum diameter of its connected components. We show that almost surely the diameter of random graph G(n, p) is close to if np → ∞. Moreover if , then the diameter of G(n, p) is concentrated on two values. In general, if , the diameter is concentrated on at most 21/c₀ + 4 values. We also proved that the diameter of G(n, p) is almost surely equal to the diameter of its giant component if np > 3.6.     

Periodic Schur functions and slit discs

A simply connected domain is called a slit disc if minus a finite number of closed radial slits not reaching the origin. A slit disc is called rational (rationally placed) if the lengths of all its circular arcs between neighboring slits (the arguments of the slits) are rational multiples of 2π. The conformal mapping of onto , (0)=0, ^′(0)>0, extends to a continuous function on mapping it onto . A finite union E of closed non-intersecting arcs e_k on is called rational if for every k, ν_E(e_k) being the harmonic measures of e_k at ∞ for the domain . A compact E is rational if and only if there is a rational slit disc such that . A compact E essentially supports a measure with periodic Verblunsky parameters if and only if for a rationally placed . For any tuple (α₁,…,α_g+1) of positive numbers with ∑_kα_k=1 there is a finite family of closed non-intersecting arcs e_k on such that ν_E(e_k)=α_k. For any set and any >0 there is a rationally placed compact such that the Lebesgue measure |EE^*| of the symmetric difference EE^* is smaller than .     

Weighted Uncertainty Principles in L

We determine sufficient conditions on positive weights W and V such that there exists continuous, strictly increasing functions Φ and Ψ on [0, ∞) such that Φ(0)=0=Ψ(0) and whenever f: R→R is a continuous integrable function. We also give an example that shows the optimality of our conditions.     

On Weighted Mean Convergence of Lagrange Interpolation for General Arrays

For n1, let {x_jn}_j=1ⁿ be n distinct points and let L_n[·] denote the corresponding Lagrange Interpolation operator. Let W : →[0,∞). What conditions on the array {x_jn}_{1jn, n1} ensure the existence of p>0 such

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for every continuous f : → with suitably restricted growth, and some “weighting factor” φ^b? We obtain a necessary and sufficient condition for such a p to exist. The result is the weighted analogue of our earlier work for interpolation arrays contained in a compact set.     

Continuity envelopes of spaces of generalised smoothness, entropy and approximation numbers

We study continuity envelopes in spaces of generalised smoothness B_pq^(s,Ψ) and F_pq^(s,Ψ) and give some new characterisations for spaces B_pq^(s,Ψ). The results are applied to obtain sharp asymptotic estimates for approximation numbers of compact embeddings of type id:B_pq^(s₁,Ψ)(U)→B_∞∞^s₂(U), where and U stands for the unit ball in . In case of entropy numbers we can prove two-sided estimates.     

A note on asymptotic behavior for negative drift random walk with dependent heavy-tailed steps and its application to risk theory

In this article, the dependent steps of a negative drift random walk are modelled as a two-sided linear process Xn =-μ ∞∑j=-∞ψn-jεj, where {ε, εn; -∞＜ n ＜ ∞}is a sequence of independent, identically distributed random variables with zero mean, μ＞0 is a constant and the coefficients {ψi;-∞＜ i ＜∞} satisfy 0 ＜∞∑j=-∞|jψj| ＜∞. Under the conditions that the distribution function of |ε| has dominated variation and ε satisfies certain tail balance conditions, the asymptotic behavior of P{supn≥0(-nμ ∞∑j=-∞εjβnj) ＞ x}is discussed. Then the result is applied to ultimate ruin probability.     

On perturbations of strongly admissible prior distributions

Consider a parametric statistical model, P(dx|θ), and an improper prior distribution, ν(dθ), that together yield a (proper) formal posterior distribution, Q(dθ|x). The prior is called strongly admissible if the generalized Bayes estimator of every bounded function of θ is admissible under squared error loss. Eaton [M.L. Eaton, A statistical diptych: Admissible inferences-recurrence of symmetric Markov chains, Annals of Statistics 20 (1992) 1147–1179] used the Blyth–Stein Lemma to develop a sufficient condition, call it , for strong admissibility of ν. Our main result says that, under mild regularity conditions, if ν satisfies and g(θ) is a bounded, non-negative function, then the perturbed prior distribution g(θ)ν(dθ) also satisfies and is therefore strongly admissible. Our proof has three basic components: (i) Eaton's [M.L. Eaton, A statistical diptych: Admissible inferences-recurrence of symmetric Markov chains, Annals of Statistics 20 (1992) 1147–1179] result that the condition is equivalent to the local recurrence of the Markov chain whose transition function is R(dθ|η)=∫Q(dθ|x)P(dx|η); (ii) a new result for general state space Markov chains giving conditions under which local recurrence is equivalent to recurrence; and (iii) a new generalization of Hobert and Robert's [J.P. Hobert, C.P. Robert, Eaton's Markov chain, its conjugate partner and -admissibility, Annals of Statistics 27 (1999) 361–373] result that says Eaton's Markov chain is recurrent if and only if the chain with transition function is recurrent. One important application of our results involves the construction of strongly admissible prior distributions for estimation problems with restricted parameter spaces.     

Galois Reconstruction of Finite Quantum Groups

Let be a (small) category and let F:  →  alg_f be a functor, where alg_f is the category of finite-dimensional measured algebras over a field k (or Frobenius algebras). We construct a universal Hopf algebra A_aut(F) such that F factorizes through a functor :  →  coalg_f(A_aut(F)), where coalg_f(A_aut(F)) is the category of finite-dimensional measured A_aut(F)-comodule algebras. This general reconstruction result allows us to recapture a finite-dimensional Hopf algebra A from the category coalg_f(A) and the forgetful functor ω: coalg_f(A) →  alg_f: we have A  A_aut(ω). Our universal construction is also done in a C*-algebra framework, and we get compact quantum groups in the sense of Woronowicz.     

Norms Concerning Subdivision Sequences and Their Applications in Wavelets

Let a:=(a(α))_{α ^s} be a finitely supported sequence of r×r matrices and M be a dilation matrix. The subdivision sequence {(a_n(α))_{α ^s}:n } is defined by a₁=a and

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Let 1≤p≤∞ and f=(f¹,…,f^r)^T be a vector of compactly supported functions in L_p( ^s). The stability is not assumed for f. The purpose of this paper is to give a formula for the asymptotic behavior of the L_p-norms of the combinations of the shifts of f with the subdivision sequence coefficients: Such an asymptotic behavior plays an essential role in the investigation of wavelets and subdivision schemes. In this paper we show some applications in the convergence of cascade algorithms, construction of inhomogeneous multiresolution analyzes, and smoothness analysis of refinable functions. Some examples are provided to illustrate the method.     

Walsh-Type Wavelet Packet Expansions

We consider a family of basic nonstationary wavelet packets generated using the Haar filters except for a finite number of scales where we allow the use of arbitrary filters. Such a system, which we call a system of Walsh-type wavelet packets, can be considered as a smooth generalization of the Walsh functions. We show that the basic Walsh-type wavelet packets share a number of metric properties with the Walsh system. We prove that the system constitutes a Schauder basis for L^p( ), 1<p<∞, and we construct an explicit function in L¹( ) for which the expansion fails. Then we prove that expansions of L^p( )-functions, 1<p<∞, in the Walsh-type wavelet packets converge pointwise a.e. Finally, we prove that the analogous results are true for periodic Walsh-type wavelet packets in L^p[0,1).     

Asymptotic Properties of Zeros of Hypergeometric Polynomials

In a paper by K. Driver and P. Duren (1999, Numer. Algorithms21, 147–156) a theorem of Borwein and Chen was used to show that for each k the zeros of the hypergeometric polynomials F(−n, kn+1; kn+2; z) cluster on the loop of the lemniscate {z: |z^k(1−z)|=k^k/(k+1)^k+1}, with Re{z}>k/(k+1) as n→∞. We now supply a direct proof which generalizes this result to arbitrary k>0, while showing that every point of the curve is a cluster point of zeros. Examples generated by computer graphics suggest some finer asymptotic properties of the zeros.     

On Banach function M-spaces

Let ω be a measure space and L_∞ be the Banach algebra of all essentially bounded measurable scalar functions defined on ω. In the present paper we study so-called Banach M-spaces X such as Banach L_∞-modules of ^m-valued measurable functions in the vector-valued case m > 1. We present with complete proofs the new technical tool via so-called representative families (of m measurable functions) for X which is necessary for constructing correctly the theory of Banach M-spaces X of ^m-valued functions with m > 1. We emphasize main new moments in proving basic theorems for Banach M-spaces of ^m-valued functions with m > 1.