Local geometry of L 1 ν L ∞ and L 1+L ∞

Extreme points of the unit sphere S (L ¹+L ^∞ ) of LL ¹+L ^∞ under the classical norm used in the interpolation theory were characterized in [8] and [11], while extreme points of S(L ¹ ∩ L ^∞) under the classical norm were characterized in [7]. In this paper extreme points of the unit sphere of L ¹+L ^∞ and L ¹ ∩ L ^∞ under the “dual” norms are characterized. Moreover, all the extreme points in L ¹ ∩ L ^∞ and L ¹+L ^∞ (under both kinds of norms) are examined if they are exposed, H-points, or strongly exposed. Smooth points in both these spaces for both the norms are also characterized. Finally, it is proved that in general the spaces L ^p +L ^q and L ^p ∩ L ^q are not isometric to Orlicz spaces, provided that 1<p<q<+∞. The paper was written while the first named author was visiting The University of Memphis The third named author is supported by KBN-Grant 2 PO3A 050 09.     

L2-CONVERGENCE AND L2-CAUCHY DOUBLE SEQUENCES

Abstract In this article, we prove a decomposition theorem for L2-convergent double sequences and introduce the notions of L2-Cauchy and L2＊-Cauchy double sequence, and then study their certain properties. Finally, we introduce the notions of regularly （L2,L）- convergence and （L2, L）-Cauchy double sequence.     

Sous-espaces de L

A typical result of the paper isTheorem. LetE be a reflexive subspace ofL ¹ (Ω, A, P) [(Ω,A, P) a probability space]. IfE contains a subspace isomorphic to l^p then for every ε > 0,E contains a subspace (1 + ε) isomorphic to l^p. The technics are probability theory and ultraproducts.     

Klassische Lösungen sind auch schwache Lösungen

Ohne Zusammenfassung     

L—模糊关系

Ｌ—模糊集与模糊集相比，在值域上使用的是更为一般的代数系统，本文以Ｌ—集合套为工具，对Ｌ—模糊映射，Ｌ—模糊变换，Ｌ—模糊关系以及它们之间的关系作了详尽的研究，得到了一些有意义的结果，这些结论可望得到广泛的应用。     

О наилучшем приближении периодических функций тригонометрическими полиномами и точных эначениях поперечников функциональных классов в L 2

Exact inequalities of Jackson-Stechkin type have been obtained for the average moduli of continuity of mth order (m ?? ?) with the weight function. The exact values of any n-widths were calculated for the function classes, which are defined by the majorant and the indicated average magnitudes.     

Autocorrelation Functions as Translation Invariants in L1 and L2

The Adler-Konheim theorem [Proc. Amer. Math. Soc. 13 (1962), 425-428] states that the collection of nth-order autocorrelation functions is a complete set of translation invariants for real-valued L¹ functions on a locally compact abelian group. It is shown here that there are proper subsets of that also form a complete set of translation invariants, and these subsets are characterized. Specifically, a subset is complete if and only if it contains infinitely many even-order autocorrelation functions. In addition, any infinite subset of is complete up to a sign. While stated here for functions on the proofs presented hold for functions on any locally compact abelian group that is not compact, in particular, on and the integer lattice      

The completeness characterization of $$C(\mathcal {L})$$ C ( L ) , $$\mathcal {L}$$ L a locale

Positivity - The result of the title is: an archimedean $$\ell $$ -group with weak unit A is (isomorphic to) $$C(\mathcal {L})$$ for some (identifiable) locale $$\mathcal {L}$$ (or, $$\mathbb...     

A Nonstandard Levy-Khintchine Formula and Levy Processes

We use methods from nonstandard analysis to obtain a short and simple derivation of the Levy-Khintchine formula via an explicit construction of certain laws of the infinitesimal increments. Consequently, any arbitrary Levy process is representable as the standard part of a hyperfinite sum of infinitesimal increments.     

Löwner expansions

A fundamental problem is to estimate the logarithmic coefficients of a power series with constant coefficient zero which represents a function which has distinct values at distinct points of the unit disk. A source of estimates is an expansion theorem for the Löwner equations which is obtained from a study of contractive substitutions in Hilbert spaces of analytic functions. The methods are an outgrowth of the theory of square summable power series [1]. Assume that σ_n is a given function of nonnegative integers n, with nonnegative values, such that σ₀ = 0 and such that σ_{n ? 1} ? σ_n when n is positive. Infinite values are allowed. The underlying Hilbert space is the set $\text{G}$_σ(0) of equivalence classes of power series f(z) = ∑ a_nzⁿ with constant coefficient zero such that f(z)²_{$\text{G}$σ⁽⁰⁾} = ∑(n/σ_n)|a_n|² is finite. Equivalence of power series f(z) and g(z) means that the coefficient of zⁿ in f(z) is equal to the coefficient of zⁿ in g(z) when σ_n is finite.     

弱(L1,L2)-BLD映射的正则性

本文首先将文[1]中的BLD映射推广为弱(L1,L2)-BLD映射,并证明了如下正则性结果存在两个可积指数p1=p1(n,L1,L2)＜n＜q1=q1(n,L1,L2),使得对任意弱(L1,L2)-BLD映射,∈W1(Ω,Rn),都有,∈(Ω,Rn),即,为(L1,L2)-BLD映射.     

Coding lemmata in L

Under the assumption that there exists an elementary embedding(henceforth abbreviated as and in particular under we prove a Coding Lemma for and find certain versions of it which are equivalent to strong regularity of cardinals below . We also prove that a stronger version of the Coding Lemma holds for a stationary set of ordinals below .Mathematics Subject Classification (2000):03E55     

双重退化抛物型方程整体解的存在性和L~∞估计

文考虑双重退化抛物型方程ut=div(|u|r|u|m--2u)+A(u)带有零边界条件的初边值问题的整体解存在性,唯一性和解在t=0,∞处的L∞模估计.证明了当u0∈Lq(Ω)时,整体解u(t)满足估计‖u(t)‖∞≤C(1+t-λβ)(1+t)-β/M,‖(u(t)|r/(m-1)u(t))‖m≤C(1+t-μ)(1+t)-σ,t＞0,这里λ,μ,σ,M,β为依赖于m,q,N和r的适当正常数.