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.     

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.     

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.     

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.     

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.     

Uniqueness and stability of regional blow-up in a porous-medium equation

We study the blow-up phenomenon for the porous-medium equation in R^N, N1, u_t=Δu^m+u^m, m>1, for nonnegative, compactly supported initial data. A solution u(x,t) to this problem blows-up at a finite time . Our main result asserts that there is a finite number of points x₁,…,x_kR^N, with |x_i−x_j|2R^* for i≠j, such that Here w_*(|x|) is the unique nontrivial, nonnegative compactly supported, radially symmetric solution of the equation in R^N and R^* is the radius of its support. Moreover u(x,t) remains uniformly bounded up to its blow-up time on compact subsets of . The question becomes reduced to that of proving that the ω-limit set in the problem consists of a single point when its initial condition is nonnegative and compactly supported.     

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).     

Universal Functions on Complex General Linear Groups

In 1929, Birkhoff proved the existence of an entire function F on with the property that for any entire function f there exists a sequence {a_k} of complex numbers such that {F(ζ+a_k)} converges to f (ζ) uniformly on compact sets. Luh proved a variant of Birkhoff's theorem and the second author proved a theorem analogous to that of Luh for the multiplicative group *. In this paper extensions of the above results to the multi-dimensional case are proved. Let M(n,  ) be the set of all square matrices of degree n with complex coefficients, and let G=GL(n,  ) be the general linear group of degree n over . We denote by (G) the set of all holomorphic functions on G. Similarly, we define ( ). Let K be the (G)-hull of a compact set K in G. Finally we denote by B(G) the set of all compact subsets K of G with K=K such that there exists a holomorphic function f on M(n,  ) with f(0)(f(K)), where (f(K)) is the ( )-hull of f(K). Our main result is the following. There exists a holomorphic function F on G such that for any KB(G), for any function f holomorphic in some neighbourhood of K, and for any >0, there exists CG with max_ZK |F(CZ)−f(Z)|<.     

Pervasive Algebras of Analytic Functions

We characterize those open U in the sphere such that A(U) is complex-pervasive, and those such that Re A(U) is real-pervasive. Pervasive means, roughly, that the uniform closure on each proper closed subset of bdy U is the space of all continuous functions (to or , as the case may be).     

Hölder continuity conditions for the solvability of Dirichlet problems involving functionals with free discontinuities

The paper deals with the problem of minimizing a free discontinuity functional under Dirichlet boundary conditions. An existence result was known so far for C¹(∂Ω) boundary data û. We show here that the same result holds for ûC^0,μ(∂Ω) if and it cannot be extended to cover the case . The proof is based on some geometric measure theoretic properties, in part introduced here, which are proved a priori to hold for all the possible minimizers.     

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).     

Optimality conditions for multiple objective fractional subset programming with ( , ρ,σ,θ )-V-type-I and related non-convex functions

In this paper, we introduce a new class of generalized convex n-set functions, called ( , ρ,σ,θ)-V-Type-I and related non-convex functions, and then establish a number of parametric and semi-parametric sufficient optimality conditions for the primal problem under the aforesaid assumptions. This work partially extends an earlier work of [G.J. Zalmai, Efficiency conditions and duality models for multiobjective fractional subset programming problems with generalized ( , α, ρ, θ)-V-convex functions, Comput. Math. Appl. 43 (2002) 1489–1520] to a wider class of functions.     

Analytic eigendistributions and applications to homogeneous trees

An analytic distribution on is an element, ν, of the dual of the space of analytic functions on K. In particular, ν defines a linear functional on the polynomial ring . In this work, we study the converse problem: given a linear functional on , try to find a minimal set K such that ν extends to an analytic distribution on K. This study was motivated by the desire to generalize a result that allows the representation of functions on a homogeneous tree as integrals of z-harmonic functions oven a certain interval. A function f on a homogeneous tree T of degree q+1 is said to be z-harmonic, if μ₁f=zf, where μ₁ is the nearest neighbor averaging operator. It was proved in [Cohen, Colonna, Adv. Appl. Math. 20 (1998) 253–274] that if |f(v)|MC^|v| for constants M>0 and , then there exist z-harmonic functions k_z such that where I is the interval with endpoints . In the present paper, we study the case when the above exponential growth condition holds with , which necessitates replacing k_z(v) dz with an analytic distribution ν_v satisfying the z-harmonicity condition μ₁ν=zν. We show that to each function on the tree satisfying the above exponential growth condition there corresponds an eigendistribution on an elliptical region containing I as the interval between its foci.     

Generalized fractal dimensions: equivalences and basic properties

Given a positive probability Borel measure μ on , we establish some basic properties of the associated functions τ_μ^±(q) and of the generalized fractal dimensions D_μ^±(q) for . We first give the equivalence of the Hentschel–Procaccia dimensions with the Rényi dimensions and the mean-q dimensions, for q>0. We then use these relations to prove some regularity properties for τ_μ^±(q) and D_μ^±(q); we also provide some estimates for these functions, in particular estimates on their behaviour at ±∞, as well as for the dimensions corresponding to convolution of two measures. We finally present some calculations for specific examples illustrating the different cases met in the article.     

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.     

Thin points for Brownian motion

Let Θ(x,r) denote the occupation measure of the ball of radius r centered at x for Brownian motion {W_t}_0≤t≤1 in . We prove that for any analytic set E in [0,1], we have

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, where dim_P(E) is the packing dimension of E. We deduce that for any a≥1, the Hausdorff dimension of the set of “thin points” x for which

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, is almost surely 2−2/a; this is the correct scaling to obtain a nondegenerate “multifractal spectrum” for the “thin” part of Brownian occupation measure. The methods of this paper differ considerably from those of our work on Brownian thick points, due to the high degree of correlation in the present case. To prove our results, we establish general criteria for determining which deterministic sets are hit by random fractals of ‘limsup type' in the presence of long-range correlations. The hitting criteria then yield lower bounds on Hausdorff dimension. This refines previous work of Khoshnevisan, Xiao and the second author, that required decay of correlations.     

Enumerating Permutation Polynomials I: Permutations with Non-Maximal Degree

Let be a conjugation class of permutations of a finite field _q. We consider the function N (q) defined as the number of permutations in for which the associated permutation polynomial has degree <q−2. In 1969, Wells proved a formula for N_[3](q) where [k] denotes the conjugation class of k-cycles. We will prove formulas for N_[k](q) where k=4,5,6 and for the classes of permutations of type [2 2],[3 2],[4 2],[3 3] and [2 2 2]. Finally in the case q=2ⁿ, we will prove a formula for the classes of permutations which are product of 2-cycles.     

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.     

Mass Points of Measures and Orthogonal Polynomials on the Unit Circle

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szeg recurrences. We assume that the reflection coefficients tend to some complex number a with 0<a<1. The orthogonality measure μ then lives essentially on the arc {e^it :αt2π−α} where sin with α(0,π). Under the certain rate of convergence it was proved in (Golinskii et al. (J. Approx. Theory96 (1999), 1–32)) that μ has no mass points inside this arc. We show that this result is sharp in a sense. We also examine the case of the whole unit circle and some examples of singular continuous measures given by their reflection coefficients.     

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 ∞.