A Note on the Embedding of Subspaces of Lp into ℓNr, 0 < r < 1, r ≦ p < 2

In this paper we extend a result by Bourgain-Lindenstrauss-Milman (see [1]). We prove: Let 0 < ? < 1/2, 0< r < 1, r< p < 2. There exists a constant C = C(r,p,?) such that if X is any n-dimensional subspace of L_p(0, l), then there exists Y ? ?^N_r with d(X, Y) ≦ 1 + ?, whenever N > Cn. As an application, we obtain the following partial result: Let 0 < r < 1. There exist constants C = C(r) and C' = C' (r) such that if X is any n-dimensional subspace of L_r(0,1), then there exists Y ? ^N_r with d(X, Y) ≦ C (logn)^l/^r, whenever N ≧ C'n.     

General Function Spaces,V. (The Spaces B and F: The Case 0<p≦∞)

In this paper quasi-BANACH spaces of (generalized) distributions are considered, including spaces of BESOV type, HARDY type and SOBOLEV type. The methods are based on the real variable approach to the HARDY spaces by FEFFERMAN and STEIN, and on the modification of this approach in connection with spaces of F-type by PEETRE.     

Multipliers in Besov Spaces and in LpΩ-Spaces (The Cases 0<p≦1 and p=∞)

The paper contains sufficient conditions for multipliers in two types of quasi-BANACH spaces: (i) weighted L_p-spaces of entire analytic functions of exponential type; 0<p≦∞, (ii) Besov spaces B, where ?∞<s<∞; 0<p≦∞; 0<q≦∞.     

On Marcinkiewicz-Zygmund Type Inequalities for Irregular Knots in Lp-Spaces, 0 < p ≦ +∞

An equivalence of a discrete norm and a continuous norm of a trigonometric polynomial is proved for the case of irregular knots in L_p-spaces, where 0 < p ≦ +∞.     

The convergence of q ‐Bernstein polynomials (0 < q < 1) in the complex plane

The paper focuses at the estimates for the rate of convergence of the q ‐Bernstein polynomials (0 < q < 1) in the complex plane. In particular, a generalization of previously known results on the possibility of analytic continuation of the limit function and an elaboration of the theorem by Wang and Meng is presented (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the number of (r,r+1)‐ factors in an (r,r+1)‐factorization of a simple graph

For integers d≥0, s≥0, a (d, d+s)‐graph is a graph in which the degrees of all the vertices lie in the set {d, d+1, …, d+s}. For an integer r≥0, an (r, r+1)‐factor of a graph G is a spanning (r, r+1)‐subgraph of G. An (r, r+1)‐factorization of a graph G is the expression of G as the edge‐disjoint union of (r, r+1)‐factors. For integers r, s≥0, t≥1, let f(r, s, t) be the smallest integer such that, for each integer d≥f(r, s, t), each simple (d, d+s) ‐graph has an (r, r+1) ‐factorization with x (r, r+1) ‐factors for at least t different values of x. In this note we evaluate f(r, s, t). © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 257‐268, 2009     

Polynomial approximation inL p (0<p<1)

We prove that forf∈L _p , 0<p<1, andk a positive integer, there exists an algebraic polynomialP _n of degree ≤n such that $$\left\| {f - P_n } \right\|_p \leqslant C\omega _k^\varphi \left( {f,\frac{1}{n}} \right)_p $$ whereω _k ^? (f,t)p is the Ditzian-Totik modulus of smoothness off inL _p , andC is a constant depending only onk andp. Moreover, iff is nondecreasing andk≤2, then the polynomialP _n can also be taken to be nondecreasing.     

Excursions of a BES0 (d) and its drift term (0<d<1)

Summary LetX be a BES₀(d) (0<d<1) with canonical decompositionX=B+(d–1)H, whereB is a brownian motion andH locally of zero energy. The process (X; H) is shown to have a local time at (0; 0), and the characteristic measure of its excursions (in Itô's sense) is described. This study leads us to new determinations of the-space variable-process defined by the occupation densities ofH taken at some optional times.     

Isometries on the Quasi-Banach Spaces Lp （0〈p〈1）

We study the extension of isometries between the unit spheres of quasi-Banach spaces Lp for 0〈p〈1. We give some sufficient conditions such that an isometric mapping from the the unit sphere of Lp（μ） into that of another LP（ν） can be extended to be a linear isometry defined on the whole space.     

Embedding Subspaces of Lp into 𝓁Np, 0 < p < 1

It is shown that if X is an n – dimensional subspace of L_p, 0 < p < 1, then there exists a subspace Y of 𝓁^N_p such that d(X, Y) ≤ 1 + ε and N ≤ C(ε, p)n(log n)³.     

Factorization of operator valued Lp for 0 ≤ p < 1

In [5], it is proved that a bounded linear operator u, from a Banach space Y into an L_p(S, ν) factors through L_p1 (S, ν) for some p₁ > 1, if Y* is of finite cotype; (S, ν) is a probability space for p = 0, and any measure space for 0 < p < 1. In this paper, we generalize this result to uv, where u : Y → L_p(S, ν) and v : X → Y are linear operators such that v* is of finite Ka?in cotype. This result gives also a new proof of Grothendieck's theorem. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Convergence of (r|~)_s(b_0h_n-h_1b_(n-1))

This paper computes the Thorn map onγ2 and proves that it is represented by 2b2,0h1,2 in the ASS. The authors also compute the higher May differential of 62,0, from which it is proved thatγs(b0hn-h1bn-1) for 2≤s < p - 1 are permanent cycles in the ASS.     

Approximation by Kantorovich-Bernstein polynomials inL p(0<p<1)-metric

The purpose of the present paper is to evaluate the error of the approximation of the function f∈L₁[0,1] by Kantorovich-Bernstein polynomials in L_p-metric (0<p<1).     

Locally Complemented Subspaces and ℒ︁p-Spaces for 0 < p < 1

We develop a theory of ??_p-spaces for 0 < p < 1, basing our definition on the concept of a locally complemented subspace of a quasi-BANACH space. Among the topics we consider are the existence of basis in ??_p-spaces, and lifting and extension properties for operators. We also give a simple construction of uncountably many separable ??_p-spaces of the form ??_p(X) where X is not a ??_p-space. We also give some applications of our theory to the spaces H_p, 0 < p < 1.