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( ₊).     

On accuracy of approximation of the spectral radius by the Gelfand formula

The famous Gelfand formula ρ(A)=limsup_n→∞A^n1/n for the spectral radius of a matrix is of great importance in various mathematical constructions. Unfortunately, the range of applicability of this formula is substantially restricted by a lack of estimates for the rate of convergence of the quantities A^n1/n to ρ(A). In the paper this deficiency is made up to some extent. By using the Bochi inequalities we establish explicit computable estimates for the rate of convergence of the quantities A^n1/n to ρ(A). The obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets.     

On the Norm of the Metric Projections

LetXbe a Banach space. GivenMa subspace ofXwe denote withP_Mthe metric projection ontoM. We defineπ(X) sup{P_M: Ma proximinal subspace ofX}. In this paper we give a bound forπ(X). In particular, whenX=L_p, we obtain the inequality P_M2^|2/p−1|, for every subspaceMofL_p. We also show thatπ(X)=π(X*).     

On -convergence of Fourier series

In this paper we consider the trigonometric Fourier series with the β-general monotone coefficients. Necessary and sufficient conditions of L₁-convergence for such a series, that is f−S_n=o(1), are obtained in terms of 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}.     

An isomorphic characterization of L-spaces

We show that a sequentially (τ)-complete topological vector lattice X_τ is isomorphic to some L¹(μ), if and only if the positive cone can be written as X₊ = ₊B for some convex, (τ)-bounded, and (τ)-closed set B X₊ {0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the “ball-generated” ordering induced by the cone Y₊ = ₊ (for u >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials.     

Theorems about the attractor for incompressible non-Newtonian flow driven by external forces that are rapidly oscillating in time but have a smooth average

This paper discusses the incompressible non-Newtonian fluid with rapidly oscillating external forces g(x,t)=g(x,t,t/) possessing the average g⁰(x,t) as →0⁺, where 0<₀<1. Firstly, with assumptions (A₁)–(A₅) on the functions g(x,t,ξ) and g⁰(x,t), we prove that the Hausdorff distance between the uniform attractors and in space H, corresponding to the oscillating equations and the averaged equation, respectively, is less than O() as →0⁺. Then we establish that the Hausdorff distance between the uniform attractors and in space V is also less than O() as →0⁺. Finally, we show for each [0,₀].     

Turán type reverse Markov inequalities for compact convex sets

For a compact convex set the well-known general Markov inequality holds asserting that a polynomial p of degree n must have p^′c(K)n²p. On the other hand for polynomials in general, p^′ can be arbitrarily small as compared to p.The situation changes when we assume that the polynomials in question have all their zeroes in the convex set K. This was first investigated by Turán, who showed the lower bounds p^′(n/2)p for the unit disk D and for the unit interval I[-1,1]. Although partial results provided general lower estimates of order , as well as certain classes of domains with lower bounds of order n, it was not clear what order of magnitude the general convex domains may admit here.Here we show that for all bounded and convex domains K with nonempty interior and polynomials p with all their zeroes lying in K p^′c(K)np holds true, while p^′C(K)np occurs for any K. Actually, we determine c(K) and C(K) within a factor of absolute numerical constant.     

Global existence, uniqueness, and continuous dependence for a semilinear initial value problem

In this paper, we consider the semilinear initial value problem associated with an operator A whose spectrum lies in a sector of the complex plane and whose resolvent satisfies (z−A)⁻¹M|z|^γ for some −1<γ<0 and all z outside the sector. The properties of existence and uniqueness of global mild solutions and continuous dependence on the initial data are investigated.     

Le lemme de Schur pour les représentations orthogonales

Let σ be an orthogonal representation of a group G on a real Hilbert space. We show that σ is irreducible if and only if its commutant σ(G)' is isomorphic to , or . This result is an analogue of the classical Schur lemma for unitary representations. In both cases (orthogonal and unitary), a representation is irreducible if and only if its commutant is a field. If σ is irreducible, we show that there exists a unitary irreducible representation π of G such that the complexification σ is unitarily equivalent to π if σ(G)' , to π π̄ if σ(G)' , and to π π if σ(G)' (here π̄ denotes the contragredient representation of π). These results are classical for a finite-dimensional σ, but seem to be new in the general case.     

A vanishing theorem on Kaehler Finsler manifolds

Let M be a connected compact complex manifold endowed with a strongly pseudoconvex complex Finsler metric F. In this paper, we first define the complex horizontal Laplacian □_h and complex vertical Laplacian □_v on the holomorphic tangent bundle T^1,0M of M, and then we obtain a precise relationship among □_h,□_v and the Hodge–Laplace operator on (T^1,0M,,), where , is the induced Hermitian metric on T^1,0M by F. As an application, we prove a vanishing theorem of holomorphic p-forms on M under the condition that F is a Kaehler Finsler metric on M.     

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).     

On Mean Convergence of Lagrange Interpolation for General Arrays

For n1, let {x_jn}ⁿ_j=1 be n distinct points in a compact set K and letL_n[·] denote the corresponding Lagrange interpolation operator. Let v be a suitably restricted function on K. What conditions on the array {x_jn}_1jn, n1 ensure the existence of p>0 such that lim_n→∞ (f−L_n[f]) v_Lp(K)=0 for very continuous f: K→ ? We show that it is necessary and sufficient that there exists r>0 with sup_n1 π_nv_Lr(K) ∑ⁿ_j=1 (1/|π′_n| (x_jn))<∞. Here for n1, π_n is a polynomial of degree n having {x_jn}ⁿ_j=1 as zeros. The necessity of this condition is due to Ying Guang Shi.     

Best constants of harmonic approximation on classes associated with the Laplace operator

We compute the best constants of approximation by entire functions of spherical type and by trigonometric polynomials of spherical degree on classes of functions f satisfying the condition Δ^kf_Lp1, where p=1 or 2 and Δ is the Laplace operator.     

Point spectra of partially power-bounded operators

Let T be an operator on a separable Banach space, and denote by σ_p(T) its point spectrum. We answer a question left open in (Israel J. Math. 146 (2005) 93–110) by showing that it is possible that be uncountable, yet Tⁿ∞. We further investigate the relationship between the growth of sequences (n_k) such that sup_kT^n_k<∞ and the possible size of .Analogous results are also derived for continuous operator semigroups (T_t)_t0.     

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.     

Strong characterizing sequences for subgroups of compact groups

In [A. Biró, V.T. Sós, Strong characterizing sequences in simultaneous Diophantine approximation, J. Number Theory 99 (2003) 405–414] we proved that if Γ is a subgroup of the torus R/Z generated by finitely many independent irrationals, then there is an infinite subset AZ which characterizes Γ in the sense that for γR/Z we have ∑_aAaγ<∞ if and only if γΓ. Here we consider a general compact metrizable Abelian group G instead of R/Z, and we characterize its finitely generated free subgroups Γ by subsets AG^*, where G^* is the Pontriagin dual of G. For this case we prove stronger forms of the analogue of the theorem of the above mentioned work, and we find necessary and sufficient conditions for a kind of strengthening of this statement to be true.