On approximation of a class by another class and extremal subspaces inL 1

Получена оценка (в опр еделенном смысле неу лучшаемая) наилучшего приближе ния в метрикеL ₁=L₁(0,2π) 2π-перио дических функций кла сса W^rH^ω = {f:f∈C^r,ω(f ^(r),δ) ≦ ω(δ)}, r = 0, 1, ..., (ω(δ) — выпуклый вверх мо дуль непрерывности) ф ункциями класса W ₁ ^r+v N = {?: ?^r+v?1)(t) —локально абсолютн о непрерывна, ∥?(^r+v∥L₁≦N}, v≧2. Доказано, что каждое п одпространство нече тной размерности, реализу ющее поперечник (по Колмогорову) класс а W ₁ ^r+v в L₁, обладает аналог ичным свойством относител ьно класса W^rH^ω при любом выпуклом вверх ω(δ).     

On the structural properties of the weight space L p(x),ω for 0 < p(x) < 1

The main purpose of this paper is to study the weight space L _p(x),ω for 0 < p(x) < 1 as well as the topology of this space. Embeddings between different Lebesgue spaces with variable exponent of summability are established. In particular, it is proved that the set of all linear continuous functionals over L _p(x),ω for 0 < p(x) < 1 consists only of the zero functional.     

On density of interpolation points,a Kadec-type theorem,and Saff's principle of contamination inL p-approximation

Letf ∈L _p (I) and denote byB _n,p (f) the polynomial of bestL _p-approximation tof of degreen (1<p<∞,I=[?1,1], the norm is weightedL _p-norm with an arbitrary positive weight). Extending a result proved by Saff and Shekhtman forp=2 we show that for every 1<p<∞ andf ∈L _p (I) (not a polynomial) points of sign change of the error functionf-B _n,p (f) are dense inI asn→∞.     

Labeling graphs with two distance constraints

Given a graph G and integers p,q,d₁ and d₂, with p>q, d₂>d₁?1, an L(d₁,d₂;p,q)-labeling of G is a function f:V(G)→{0,1,2,…,n} such that |f(u)−f(v)|?p if d_G(u,v)?d₁ and |f(u)−f(v)|?q if d_G(u,v)?d₂. A k-L(d₁,d₂;p,q)-labeling is an L(d₁,d₂;p,q)-labeling f such that max_v∈V(G)f(v)?k. The L(d₁,d₂;p,q)-labeling number ofG, denoted by , is the smallest number k such that G has a k-L(d₁,d₂;p,q)-labeling. In this paper, we give upper bounds and lower bounds of the L(d₁,d₂;p,q)-labeling number for general graphs and some special graphs. We also discuss the L(d₁,d₂;p,q)-labeling number of G, when G is a path, a power of a path, or Cartesian product of two paths.     

Widths of classes of periodic differentiable functions in the space L 2[0, 2π]

We obtain exact values of different n-widths for classes of differentiable periodic functions in the space L ₂[0, 2π] satisfying the constraint $$ \left( {\int_0^h {\omega _m^p \left( {f^{\left( r \right)} ;t} \right)dt} } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \leqslant \Phi \left( h \right) $$ , where 0 < h < ∞, 1/r < p ≤ 2, r ∈ ?, and ω _m (f ^(r); t) is the modulus of continuity of mth order of the derivative f ^(r)(x) ∈ L ₂[0, 2π].     

Convolution on L p -spaces of a locally compact group

We have recently shown that, for 2 < p < ∞, a locally compact group G is compact if and only if the convolution multiplication f * g exists for all f, g ∈ L ^p (G). Here, we study the existence of f * g for all f, g ∈ L ^p (G) in the case where 0 < p ≤ 2. Also, for 0 < p < ∞, we offer some necessary and sufficient conditions for L ^p (G) * L ^p (G) to be contained in certain function spaces on G.     

On Bernstein's inequality for entire functions of exponential type

It was shown by S.N. Bernstein that if f is an entire function of exponential type τ such that |f(x)|?M for −∞<x<∞, then |f^′(x)|?Mτ for −∞<x<∞. If p is a polynomial of degree at most n with |p(z)|?M for |z|=1, then f(z):=p(eiz) is an entire function of exponential type n with |f(x)|?M on the real axis. Hence, by the just mentioned inequality for functions of exponential type, |p^′(z)|?Mn for |z|=1. Lately, many papers have been written on polynomials p that satisfy the condition znp(1/z)≡p(z). They do form an intriguing class. If a polynomial p satisfies this condition, then f(z):=p(eiz) is an entire function of exponential type n that satisfies the condition f(z)≡einzf(−z). This led Govil [N.K. Govil, Lp inequalities for entire functions of exponential type, Math. Inequal. Appl. 6 (2003) 445-452] to consider entire functions f of exponential type satisfying f(z)≡eiτzf(−z) and find estimates for their derivatives. In the present paper we present some additional observations about such functions.     

Matrix refinement type equations

Taking advantage of perpetuities and the asymptotic behavior of products of random matrices we obtain the direct form of the Fourier transform of an L¹-solution of the following random matrix refinement type equation

f(x)=_∫Ω|detL(ω)|C(ω)f(L(ω)x-M(ω))P(dω),     

A Leibniz differentiation formula for positive operators

Is is shown that for n→+∞ the Leibnizian combination L′_n(fg)−fL′_n(g)−gL′_n(f) converges uniformly to zero on a compact interval W if the positive operators L_n belong to a certain class (including Bernstein, Gauss-Weierstrass and many others), and if the moduli of continuity of f,g satisfy ω_W(f;h)ω_W(g;h)=o(h) as h→0+. A counterexample shows that Lipschitz conditions are not appropriate to bring about a second-order version of this formula.     

Graph maps whose periodic points form a closed set

A continuous map f from a graph G to itself is called a graph map. Denote by P(f), R(f), ω(f), Ω(f) and CR(f) the sets of periodic points, recurrent points, ω-limit points, non-wandering points and chain recurrent points of f respectively. It is well known that P(f)⊂R(f)⊂ω(f)⊂Ω(f)⊂CR(f). Block and Franke (1983) [5] proved that if f:I→I is an interval map and P(f) is a closed set, then CR(f)=P(f). In this paper we show that if f:G→G is a graph map and P(f) is a closed set, then ω(f)=R(f). We also give an example to show that, for general graph maps f with P(f) being a closed set, the conclusion ω(f)=R(f) cannot be strengthened to Ω(f)=R(f) or ω(f)=P(f).     

Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series

A generalization of Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series is investigated with the help of a continuous function θ. Under some weak conditions on θ we show that the maximal operator of the Marcinkiewicz-θ-means of a tempered distribution is bounded from Hp(Xd) to Lp(Xd) for all d/(d+α)<p?∞ and, consequently, is of weak type (1,1), where 0<α?1 is depending only on θ and X=R or X=T. As a consequence we obtain a generalization of a summability result due to Marcinkiewicz and Zhizhiashvili for d-dimensional Fourier transforms and Fourier series, more exactly, the Marcinkiewicz-θ-means of a function f∈L₁(Xd) converge a.e. to f. Moreover, we prove that the Marcinkiewicz-θ-means are uniformly bounded on the spaces Hp(Xd) and so they converge in norm (d/(d+α)<p<∞). Similar results are shown for conjugate functions. Some special cases of the Marcinkiewicz-θ-summation are considered, such as the Fejér, Cesàro, Weierstrass, Picar, Bessel, de La Vallée-Poussin, Rogosinski and Riesz summations.     

Sufficient conditions for the Lebesgue integrability of Fourier transforms

We give sufficient conditions for the Lebesgue integrability of the Fourier transform of a function f ∈ L ^p (?) for some 1 < p ≤ 2. These sufficient conditions are in terms of the L ^p integral modulus of continuity of f; in particular, they apply for functions in the integral Lipschitz class Lip(α, p) and for functions of bounded s-variation for some 0 < s < p. Our theorems are nonperiodic versions of the classical theorems of Bernstein, Szász, Zygmund and Salem, and recent theorems of Gogoladze and Meskhia on the absolute convergence of Fourier series.     

Hankel and Berezin Type Operators on Weighted Besov Spaces of Holomorphic Functions on Polydisks

Let S be the space of functions of regular variation and let ω = (ω₁,..., ω_n), ω_j ∈ S. The weighted Besov space of holomorphic functions on polydisks, denoted by B _p (ω) (0 < p < +∞), is defined to be the class of all holomorphic functions f defined on the polydisk U ⁿ such that \(||f||_{{B_{P(\omega )}}}^P = \int_{{U^n}} {|Df(z){|^p}\prod\limits_{j = 1}^n {{\omega _j}{{(1 - |{z_j}{|^2})}^{P - 2}}dm{a_{2n}}(z) < \infty } } \), where dm_2n(z) is the 2ndimensional Lebesgue measure on U ⁿ and D stands for a special fractional derivative of f.We prove some theorems concerning boundedness of the generalized little Hankel and Berezin type operators on the spaces B _p (ω) and L _p (ω) (the weighted L _p -space).     

Basic Fourier series on a q-linear grid: Convergence theorems

For 0<q<1 define the symmetric q-linear operator acting on a suitable function f(x) by δf(x)=f(q1/2x)−f(q−1/2x). The q-linear initial value problem , f(0)=1, has two entire functions Cq(z) and Sq(z) as linearly independent solutions, which are orthogonal on a discrete set. Sufficient conditions for pointwise convergence and for uniform convergence of the corresponding Fourier expansion are given.     

Functions of operators under perturbations of class Sp

This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λα(R) and 1<p<∞, the operator f(A)−f(B) belongs to Sp/α, whenever A and B are self-adjoint operators with A−B∈Sp. We also obtain sharp estimates for the Schatten-von Neumann norms ‖f(A)−f(B)Sp/α_‖ in terms of ‖A−BSp_‖ and establish similar results for other operator ideals. We also estimate Schatten-von Neumann norms of higher order differences . We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f(A)−f(B) to belong to Sq under the assumption that A−B∈Sp. We also obtain Schatten-von Neumann estimates for quasicommutators f(A)R−Rf(B), and introduce a spectral shift function and find a trace formula for operators of the form f(A−K)−2f(A)+f(A+K).     

Semilinear elliptic problems near resonance with a nonprincipal eigenvalue

We consider the Dirichlet problem for the equation −Δu=λu±f(x,u)+h(x) in a bounded domain, where f has a sublinear growth and h∈L². We find suitable conditions on f and h in order to have at least two solutions for λ near to an eigenvalue of −Δ. A typical example to which our results apply is when f(x,u) behaves at infinity like a(x)|u|^q−2u, with M>a(x)>δ>0, and 1<q<2.     

Labeling bipartite permutation graphs with a condition at distance two

An L(p,q)-labeling of a graph G is an assignment f from vertices of G to the set of non-negative integers {0,1,…,λ} such that |f(u)−f(v)|≥p if u and v are adjacent, and |f(u)−f(v)|≥q if u and v are at distance 2 apart. The minimum value of λ for which G has L(p,q)-labeling is denoted by λ_p,q(G). The L(p,q)-labeling problem is related to the channel assignment problem for wireless networks.In this paper, we present a polynomial time algorithm for computing L(p,q)-labeling of a bipartite permutation graph G such that the largest label is at most (2p−1)+q(bc(G)−2), where bc(G) is the biclique number of G. Since λ_p,q(G)≥p+q(bc(G)−2) for any bipartite graph G, the upper bound is at most p−1 far from optimal.     

The ω-limit set of a graph map

Let G be a graph and be continuous. Denote by P(f), , ω(f) and Ω(f) the set of periodic points, the closure of the set of periodic points, ω-limit set and non-wandering set of f, respectively. In this paper we show that: (1) v∈ω(f) if and only if v∈P(f) or there exists an open arc L=(v,w) contained in some edge of G such that every open arc U=(v,c)⊂L contains at least 2 points of some trajectory; (2) v∈ω(f) if and only if every open neighborhood of v contains at least r+1 points of some trajectory, where r is the valence of v; (3) ; (4) if , then x has an infinite orbit.     

On S-asymptotically ω-periodic functions on Banach spaces and applications

This paper is devoted to the study of the class of continuous and bounded functions for which exists ω>0 such that limt→∞(f(t+ω)−f(t))=0 (in the sequel called S-asymptotically ω-periodic functions). We discuss qualitative properties and establish some relationships between this type of functions and the class of asymptotically ω-periodic functions. We also study the existence of S-asymptotically ω-periodic mild solutions of the first-order abstract Cauchy problem in Banach spaces.