Estimation of dimension functions of band-limited wavelets

The dimension function D_ψ of a band-limited wavelet ψ is bounded by n if its Fourier transform is supported in [−(2ⁿ⁺²/3)π,(2ⁿ⁺²/3)π]. For each and for each , 0<<δ=δ(n), we construct a wavelet ψ with supp

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such that D_ψ>n on a set of positive measure, which proves that [−(2ⁿ⁺²/3)π,(2ⁿ⁺²/3)π] is the largest symmetric interval for estimating the dimension function by n. This construction also provides a family of (uncountably many) wavelet sets each consisting of infinite number of intervals.     

Julia Sets of Certain Exponential Functions

We characterize the Julia sets of certain exponential functions. We show that the Julia sets J(F_λn) of F_λn(z) = λ_ne^zn where λ_n > 0 is the whole plane , provided that lim_k → ∞ F^k_λn(0) = ∞. In particular, this is true when λ_n are real numbers such that . On the other hand, if , then J(F_λn) is nowhere dense in and is the complement of the basin of attraction of the unique real attractive fixed point of F_λn. We then prove similar results for the functions[formula] where λ_i    − {0}, 1 ≤ i ≤ n + 1, a_j > 1, 1 ≤ j ≤ n, and m, n ≥ 1.     

Classification Theorems for General Orthogonal Polynomials on the Unit Circle

The set of all probability measures σ on the unit circle splits into three disjoint subsets depending on properties of the derived set of {|_n|²dσ}_n0, denoted by Lim(σ). Here {_n}_n0 are orthogonal polynomials in L²(dσ). The first subset is the set of Rakhmanov measures, i.e., of σ with {m}=Lim(σ), m being the normalized (m( )=1) Lebesgue measure on . The second subset Mar( ) consists of Markoff measures, i.e., of σ with mLim(σ), and is in fact the subject of study for the present paper. A measure σ, belongs to Mar( ) iff there are >0 and l>0 such that sup{|a_n+j|:0jl}>, n=0,1,2,…,{a_n} is the Geronimus parameters (=reflectioncoefficients) of σ. We use this equivalence to describe the asymptotic behavior of the zeros of the corresponding orthogonal polynomials (see Theorem G). The third subset consists of σ with {m}Lim(σ). We show that σ is ratio asymptotic iff either σ is a Rakhmanov measure or σ satisfies the López condition (which implies σMar( )). Measures σ satisfying Lim(σ)={ν} (i.e., weakly asymptotic measures) are also classified. Either ν is the sum of equal point masses placed at the roots of zⁿ=λ, λ , n=1,2,…, or ν is the equilibrium measure (with respect to the logarithmic kernel) for the inverse image under an m-preserving endomorphism z→zⁿ, n=1,2,…, of a closed arc J (including J= ) with removed open concentric arc J₀ (including J₀=). Next, weakly asymptotic measures are completely described in terms of their Geronimus parameters. Finally, we obtain explicit formulae for the parameters of the equilibrium measures ν and show that these measures satisfy {ν}=Lim(ν).     

The Linear Part of a Discontinuously Acting Euclidean Semigroup

Let ⁿ be a Euclidean space and let S be a Euclidean semigroup, i.e., a subsemigroup of the group of isometries of ⁿ. We say that a semigroup S acts discontinuously on ⁿ if the subset {s  S:sK ∩ K ≠ } is finite for any compact set K of ⁿ. The main results of this work areTheorem.If S is a Euclidean semigroup which acts discontinuously on ⁿ, then the connected component of the closure of the linear part ℓ(S) of S is a reducible group.Corollary.Let S be a Euclidean semigroup acting discontinuously on ⁿ; then the linear part ℓ(S) of S is not dense in the orthogonal group O(n).These results are the first step in the proof of the followingMargulis' Conjecture.If S is a crystallographic Euclidean semigroup, then S is a group.     

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.     

Boundedness of generalized higher commutators of Marcinkiewicz integrals

Let (b) ＝ (b1,…,bm) be a finite family of locally integrable functions. Then,we introduce generalized higher commutator of Marcinkiwicz integral as follows:μ(b)Ω=(∫∞o|F(b)Ω,t(f)(x)|2et/t)1/2,whereF(b)Ω(f)(x)＝1/t∫|x-y|≤tΩ(x-y)/|x-y|n-1m∏j=1(bj(x)-bj(y))f(y)dy.When bj ∈(A)βj, 1≤j≤m, 0＜βj＜1,m∑j=1βj =β＜n, and Ω is homogeneous of degree zero and satisfies the cancelation condition, we prove that μ(b)Ω is bounded from Lp(Rn)to Ls(Rn), where 1 ＜ p ＜ n/β and 1/s = 1/p -β/n. Moreover, if Ω also satisfies some Lq-Dini condition, then μ(b)Ω is bounded from Lp(Rn) to (F)β,∞p(Rn) and on certain Hardy spaces. The article extends some known results.     

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.     

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.     

On a Conjecture of Ditzian and Runovskii

Let M_θ be the mean operator on the unit sphere in ⁿ, n3, which is an analogue of the Steklov operator for functions of single variable. Denote by D the Laplace–Beltrami operator on the sphere which is an analogue of second derivative for functions of single variable. Ditzian and Runovskii have a conjecture on the norm of the operator θ²D(M_θ)^m, m2 from X=L^p (1p∞) to itself which can be expressed as

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. We give a proof of this conjecture.     

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 .     

Compressing Mappings on Primitive Sequences over Z/(2) and Its Galois Extension

Let f(x) be a strongly primitive polynomial of degree n over Z/(2^e), η(x₀,x₁,…,x_e−2) a Boolean function of e−1 variables and (x₀,x₁,…,x_e−1)=x_e−1+η(x₀,x₁,…,x_e−2)G (f(x),Z/(2^e)) denotes the set of all sequences over Z/(2^e) generated by f(x), F₂^∞ the set of all sequences over the binary field F₂, then the compressing mapping

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is injective, that is, for , G(f(x),Z/(2^e)), = if and only if Φ( )=Φ( ), i.e., ( ₀,…, _e−1)=( ₀,…, _e−1) mod 2. In the second part of the paper, we generalize the above result over the Galois rings.     

An n log n Algorithm for Online BDD Refinement

Binary decision diagrams are in widespread use in verification systems for the canonical representation of finite functions. Here we consider multivalued BDDs, which represent functions of the form : ^ν →  , where is a finite set of leaves. We study a rather natural online BDD refinement problem: a partition of the leaves of several shared BDDs is gradually refined, and the equivalence of the BDDs under the current partition must be maintained in a discriminator table. We show that it can be solved in O(n log n) time if n bounds both the size of the BDDs and the total size of update operations. Our algorithm is based on an understanding of BDDs as the fixed points of an operator that in each step splits and gathers nodes. We apply our algorithm to show that automata BDD-represented transition functions can be minimized in time O(n · log n), where n is the total number of BDD nodes representing the automaton. This result is not an instance of Hopcroft's classical minimization algorithm, which breaks down for BDD-represented automata because of the BDD path compression property.     

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.     

Lower bounds for weak sense of direction

A graph with n vertices and maximum degree Δ cannot be given weak sense of direction using less than Δ colours. It is known that n colours are always sufficient, but it has been conjectured that just Δ+1 are really needed. On the contrary, we show that for sufficiently large n there are graphs requiring Δ+Ω((nloglogn)/logn) colours. Moreover, we prove that, in terms of the maximum degree, colours are necessary.     

Stability and other limit laws for exit times of random walks from a strip or a halfplane

We show that the passage time, T^*(r), of a random walk S_n above a horizontal boundary at r (r≥0) is stable (in probability) in the sense that as r→∞ for a deterministic function C(r)>0, if and only if the random walk is relatively stable in the sense that as n→∞ for a deterministic sequence B_n>0. The stability of a passage time is an important ingredient in some proofs in sequential analysis, where it arises during applications of Anscombe's Theorem. We also prove a counterpart for the almost sure stability of T^*(r), which we show is equivalent to E|X|<∞, EX>0. Similarly, counterparts for the exit of the random walk from the strip {|y|≤r} are proved. The conditions arefurther related to the relative stability of the maximal sum and the maximum modulus of the sums. Another result shows that the exit position of the random walk outside the boundaries at ±r drifts to ∞ as r→∞ if and only if the random walk drifts to ∞.     

Universal Polynomial Majorants on Convex Bodies

Let K be a convex body in ^d (d2), and denote by B_n(K) the set of all polynomials p_n in ^d of total degree n such that |p_n|1 on K. In this paper we consider the following question: does there exist a p*_nB_n(K) which majorates every element of B_n(K) outside of K? In other words can we find a minimal γ1 and p*_nB_n(K) so that |p_n(x)|γ |p*_n(x)| for every p_nB_n(K) and x ^d\K? We discuss the magnitude of γ and construct the universal majorants p*_n for evenn. It is shown that γ can be 1 only on ellipsoids. Moreover, γ=O(1) on polytopes and has at most polynomial growth with respect to n, in general, for every convex body K.     

On Acyclic Orientations and Sequential Dynamical Systems

We study a class of discrete dynamical systems that consists of the following data: (a) a finite (labeled) graph Y with vertex set {1, …, n}, where each vertex has a binary state, (b) a vertex labeled multi-set of functions (F_i, Y: ₂ⁿ →  ₂ⁿ)_i, and (c) a permutation π  S_n. The function F_i, Y updates the binary state of vertex i as a function of the states of vertex i and its Y-neighbors and leaves the states of all other vertices fixed. The permutation π represents a Y-vertex ordering according to which the functions F_i, Y are applied. By composing the functions F_i, Y in the order given by π we obtain the sequential dynamical system (SDS):

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In this paper we first establish a sharp, combinatorial upper bound on the number of non-equivalent SDSs for fixed graph Y and multi-set of functions (F_i, Y). Second, we analyze the structure of a certain class of fixed-point-free SDSs.     

Interpolatory Subdivision Schemes Induced by Box Splines

This paper is devoted to a study of interpolatory refinable functions. If a refinable function φ on ^sis continuous and fundamental, i.e., φ(0)=1 and φ(α)=0 for α ^s\{0}, then its corresponding mask bsatisfies b(0)=1 and b(2α)=0 for all α ^s\{0}. Such a refinement mask is called an interpolatory mask. We establish the existence and uniqueness of interpolatory masks which are induced by masks of box splines whose shifts are linearly independent.     

Discrete Fourier–Neumann series

Let J_μ denote the Bessel function of order μ. The systemwith n=0,1,…,α>−1, and where p_s denotes the sth positive zero of J_α(ax), is orthonormal in . In this paper, we study the mean convergence of the Fourier series with respect to this system. Also, we describe the space in which the span of the system is dense.     

Markov inequality for polynomials of degree n with m distinct zeros

Let be the collection of all polynomials of degree at most n with real coefficients that have at most m distinct complex zeros. We prove thatfor every . This is far away from what we expect. We conjecture that the Markov factor 32·8^mn above may be replaced by cmn with an absolute constant c>0. We are not able to prove this conjecture at the moment. However, we think that our result above gives the best-known Markov-type inequality for on a finite interval when mc log n.