On constants in some inequalities for intermediate derivatives on a finite interval

Let L_q (1q<∞) be the space of functions f measurable on I=[−1,1] and integrable to the power q, with normL_∞ is the space of functions measurable on I with normWe denote by AC the set of all functions absolutely continuous on I. For nN, q[1,∞] we setW_n,q={f:f⁽ⁿ⁻¹⁾AC, f⁽ⁿ⁾L_q}.In this paper, we consider the problem of accuracy of constants A, B in the inequalities (1)|| f^(m)||_qA|| f||_p+B|| f^(m+k+1)||_r, mN, kW; p,q,r[1,∞], fW_m+k+1,r.     

Inner Functions and Cyclic Composition Operators on H(Bn)

The present paper shows that the algebra generated by {C|  Aut(B_n)} is cyclic on H²(B_n), and any nonconstant function f  H²(B_n) is a cyclic vector of . In addition, the hypercyclic and cyclic composition operators will be discussed.     

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.     

Optimal Sobolev Imbeddings Involving Rearrangement-Invariant Quasinorms

Let m and n be positive integers with n2 and 1mn−1. We study rearrangement-invariant quasinorms _R and _D on functions f: (0, 1)→ such that to each bounded domain Ω in ⁿ, with Lebesgue measure |Ω|, there corresponds C=C(|Ω|)>0 for which one has the Sobolev imbedding inequality _R(u*(|Ω| t))C_D(|^mu|* (|Ω| t)), uC^m₀(Ω), involving the nonincreasing rearrangements of u and a certain mth order gradient of u. When m=1 we deal, in fact, with a closely related imbedding inequality of Talenti, in which _D need not be rearrangement-invariant, _R(u*(|Ω| t))C_D((d/dt) ∫_{{x ⁿ : |u(x)|>u*(|Ω| t)}} |(u)(x)| dx), uC¹₀(Ω). In both cases we are especially interested in when the quasinorms are optimal, in the sense that _R cannot be replaced by an essentially larger quasinorm and _D cannot be replaced by an essentially smaller one. Our results yield best possible refinements of such (limiting) Sobolev inequalities as those of Trudinger, Strichartz, Hansson, Brézis, and Wainger.     

Maximal Inequalities and Lebesgue's Differentiation Theorem for Best Approximant by Constant over Balls

For fL_p( ⁿ), with 1p<∞, >0 and x ⁿ we denote by T(f)(x) the set of every best constant approximant to f in the ball B(x, ). In this paper we extend the operators T_p to the space L_p−1( ⁿ)+L_∞( ⁿ), where L₀ is the set of every measurable functions finite almost everywhere. Moreover we consider the maximal operators associated to the operators T_p and we prove maximal inequalities for them. As a consequence of these inequalities we obtain a generalization of Lebesgue's Differentiation Theorem.     

Relative rearrangement and Lebesgue spaces with variable exponent

We apply the techniques of monotone and relative rearrangements to the nonrearrangement invariant spaces L^p()(Ω) with variable exponent. In particular, we show that the maps uL^p()(Ω)→k(t)u_*L^p_*()(0,measΩ) and uL^p()(Ω)→u_*L^p*()(0,measΩ) are locally -Hölderian (u_* (resp. p^*) is the decreasing (resp. increasing) rearrangement of u (resp. p)). The pointwise relations for the relative rearrangement are applied to derive the Sobolev embedding with eventually discontinuous exponents.     

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 Approximation Functional and Interpolation with Perturbed Continuity

The continuity conditions at the endpoints of interpolation theorems, Ta_BjM_j a_Aj for j=0, 1, can be written with the help of the approximation functional: E(t, Ta; B₁, B₀)_L^∞M₀ a_A0 and E(t, Ta; B₀, B₁)_L^∞M₁ a_A1. As a special case of the results we present here we show that in the hypotheses of the interpolation theorem the L^∞ norms can be replaced by BMO( ₊) norms. This leads to a strong version of the Stein-Weiss theorem on interpolation with change of measure. Another application of our results is that the condition fL⁰, i.e., f*L^∞, where f*(γ)=μ{|f|>γ} is the distribution function of f, can be replaced in interpolation with L(p, q) spaces by the weaker f*BMO( ₊).     

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.     

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)|<.     

On multipartite posets

A poset P=(X,) is m-partite if X has a partition X=X₁X_m such that (1) each X_i forms an antichain in P, and (2) xy implies xX_i and yX_j where i<j. In this article we derive a tight asymptotic upper bound on the order dimension of m-partite posets in terms of m and their bipartite sub-posets in a constructive and elementary way.     

Optimal Recovery of the Derivative of Periodic Analytic Functions from Hardy Classes

LetS_β{z : |Im z|<β}. For 2π-periodic functions which are analytic inS_βwithp-integrable boundary values, we construct an optimal method of recovery off′(ξ), ξS_β, using information about the valuesf(x₁), mldr;, f(x_n), x_j[0, 2π).     

Strong and uniform mean stability of cosine and sine operator functions

It is first observed that a uniformly bounded cosine operator function C() and the associated sine function S() are totally non-stable. Then, using a zero-one law for the Abel limit of a closed linear operator, we prove some results concerning strong mean stability and uniform mean stability of C(). Among them are: (1) C() is strongly (C,1)-mean stable (or (C,2)-mean stable, or Abel-mean stable) if and only if 0ρ(A)σ_c(A); (2) C() is uniformly (C,2)-mean stable if and only if S() is uniformly (C,1)-mean stable, if and only if , if and only if , if and only if C() is uniformly Abel-mean stable, if and only if S() is uniformly Abel-mean stable, if and only if 0ρ(A).     

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.     

LOCAL TIME ANALYSIS OF ADDITIVE LEVY PROCESSES WITH DIFFERENT L(E)VY EXPONENTS

Let X1 XN be independent, classical Levy processes on R^d with Levy exponents ψ1,…, ψN, respectively. The corresponding additive Levy process is defined as the following N-parameter random field on R^d, X（t） △= X1（t1） ＋ ... ＋ XN（tN）, At∈N. Under mild regularity conditions on the ψi＇s, we derive estimate for the local and uniform moduli of continuity of local times of X = {X（t）; t ∈R^N}.     

Regularity results for stable-like operators

For α[1,2) we consider operators of the form

and for α(0,1) we consider the same operator but where the f term is omitted. We prove, under appropriate conditions on A(x,h), that any solution u to will be in C^α+β if fC^β.     

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

Full-size image

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.     

Convergence of a Halpern-type iteration algorithm for a class of pseudo-contractive mappings

Let E be a real reflexive Banach space with uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed and convex subset of E. Let T:K→K be a strictly pseudo-contractive map and let L>0 denote its Lipschitz constant. Assume F(T){xK:Tx=x}≠0/ and let zF(T). Fix δ(0,1) and let δ^* be such that δ^*δL(0,1). Define , where δ_n(0,1) and limδ_n=0. Let {α_n} be a real sequence in (0,1) which satisfies the following conditions: . For arbitrary x₀,uK, define a sequence {x_n}K by x_n+1=α_nu+(1−α_n)S_nx_n. Then, {x_n} converges strongly to a fixed point of T.