Finitely purely atomic measures: Coincidence and rigidity properties

This paper is a natural continuation of the paper [2] by the same author. We shall prove that several coincidence and rigidity phenomena which usually do not appear are possible only in case the underlying measure space is trivial (i.e. is a finite union of atoms). Examples: coincidence of twoL ^p spaces, reflexivity ofL ¹, Radon—Nikodym property ofL ^∞, coincidence of Dunford, Pettis or Bochner integrability, coincidence of theL ^p space and of the weakL ^p space.     

Test elements for endomorphisms of free groups and algebras

LetE be a bounded Borel subset of ℝⁿ,n≥2, of positive Lebesgue measure andP _E the corresponding ‘Pompeiu transform”. We prove thatP _E is injective onL ^p(ℝⁿ) if 1≤p≤2n/(n-1). We explore the connection between this problem and a Wiener-Tauberian type theorem for theM(n) action onL ^q(ℝⁿ) for various values ofq. We also take up the question of whenP _E is injective in caseE is of finite, positive measure, but is not necessarily a bounded set. Finally, we briefly look at these questions in the contexts of symmetric spaces of compact and non-compact type.     

On the Orlicz function spacesL M (0, ∞)

The isomorphic properties of the Orlicz function spacesL _M (0, ∞) are investigated. Especially we treat the question, whether theL _p-spaces are the only symmetric function spaces on (0, ∞), which are isomorphic to a symmetric function space on (0, 1). For the class of slowly varying Orlicz functions we answer this in the affirmative, and we also prove some results concerning the general case, which indicate, that it might be true there also.     

Rates of convex approximation in non-hilbert spaces

This paper deals with sparse approximations by means of convex combinations of elements from a predetermined “basis” subsetS of a function space. Specifically, the focus is on therate at which the lowest achievable error can be reduced as larger subsets ofS are allowed when constructing an approximant. The new results extend those given for Hilbert spaces by Jones and Barron, including, in particular, a computationally attractive incremental approximation scheme. Bounds are derived for broad classes of Banach spaces; in particular, forL _p spaces with 1<p<∞, theO (n ^−1/2) bounds of Barron and Jones are recovered whenp=2. One motivation for the questions studied here arises from the area of “artificial neural networks,” where the problem can be stated in terms of the growth in the number of “neurons” (the elements ofS) needed in order to achieve a desired error rate. The focus on non-Hilbert spaces is due to the desire to understand approximation in the more “robust” (resistant to exemplar noise)L _p, 1 ≤p<2, norms. The techniques used borrow from results regarding moduli of smoothness in functional analysis as well as from the theory of stochastic processes on function spaces.     

On the concentration of eigenvalues of random symmetric matrices

It is shown that for every 1≤s≤n, the probability that thes-th largest eigenvalue of a random symmetricn-by-n matrix with independent random entries of absolute value at most 1 deviates from its median by more thant is at most 4e ⁻ t ²32 ^s2. The main ingredient in the proof is Talagrand’s Inequality for concentration of measure in product spaces. Research supported in part by a USA — Israel BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Research supported in part by a USA — Israel BSF grant and by a Bergmann Memorial Grant.     

SeparableG spaces are isomorphic toC(K) spaces

We show that a wide class of separable preduals ofL ¹(μ) spaces, namely, theG spaces, introduced by Grothendieck, are isomorphic toC(K) spaces. This is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor J. Lindenstrauss. I wish to thank Professor Lindenstrauss for his interest and advice.     

On the stability of the completeness of L α spaces

The object of this paper is to study the stability of the quasi-completeness of theL ^α spaces for locally convex vector valued functions, by the inductive limits and the countable products.     

A substitute for the Sard-Smale theorem in theC 1 case

IfF is a Fredholm mapping of indexΝ ∃ ℤ and classC ^max(Ν,0)+1 between separable Banach spaces, the Sard—Smale theorem yields the existence of arbitrarily small perturbations ofF having 0 as a regular value. The smoothness requirement cannot be weakened in the Sard—Smale theorem itself, at least whenΝ ≥ 0, but we prove that the approximation result remains valid irrespective of the indexΝ whenF is only of classC ¹ and satisfies appropriate properness-like conditions. The separability of the spaces is not needed either. Everything carries over to the setting of Banach manifolds modeled on spaces with a norm of classC ¹ away from the origin. We also show that in Banach spaces, theC ¹ norm assumption can be dropped without major prejudice. The application to degree theory forC ¹ Fredholm mappings of index 0 is developed in a separate paper.     

Sharp Estimates in Bergman and Besov Spaces on Bounded Symmetric Domains

Sharp estimates of the point-evaluation functional in weighted Bergman spaces L ^p _a (Ω, dν _α) and for the point-evaluation derivalive functional in Besov spaces B ^p (Ω) are obtained for bounded symmetric domains Ω in ℂ ⁿ . Received October 25, 1999, Accepted December 6, 2000     

Permutations of the Walsh system that preserve Lebesgue constants

A class of permutations of the Walsh-Paley system that preserve the lebesgue constants and theL ^p -norms of the Dirichlet kernel is distinguished. Thus it is proved, in particular, that Fine’s estimates and calculations of the Lebesgue constants for the Walsh—Paley system hold for the Walsh systems in the enumerations of Walsh and Kaczmarz. A third algorithm for calculating the Lebesgue constants, which is different from those obtained by Fine and which also makes it possible to calculate theL ^p -norms of the Dirichlet kernels, is proposed. It is shown that not all permutations of the Walsh system even within the blocks preserve the Lebesgue constants. The distinguished class of permutations includes theTW-systems of Schipp, which are not, in general, permutations within the blocks. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 36–48, July, 2000.     

Some geometrical properties of Banach spaces of polynomials

We investigate the asymmetry, gl constants and best factorization estimates of then-dimensional spaces of polynomialsH _p ⁿ =span{e ^ikx;k=1,2,…,n} equipped with theL _p norm for 1≦p≦∞. Supported in part by NSF grant # MCS-8109561.     

Hardy spaces for non-compactly causal symmetric spaces and the most continuous spectrum

Let G/H be a semisimple symmetric space. Then the space L²(G/H) can be decomposed into a finite sum of series of representations induced from parabolic subgroups of G. The most continuous part of the spectrum of L²(G/H) is the part induced from the smallest possible parabolic subgroup. In this paper we introduce Hardy spaces canonically related to this part of the spectrum for a class of non-compactly causal symmetric spaces. The Hardy space is a reproducing Hilbert space of holomorphic functions on a bounded symmetric domain of tube type, containing G/H as a boundary component. A boundary value map is constructed and we show that it induces a G-isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part L_mc²(G/H) of L²(G/H). We also relate our Hardy space to the classical Hardy space on the bounded symmetric domain.Supported in part by NSF-grant DMS-0070816 and the MSRISupported in part by NSF-grant DMS-0097314 and the MSRISupported in part by NSF-grant DMS-0070607 and the MSRI     

M-structure in dual Banach spaces

A result previously known only for certain ordered Banach spaces is generalized to arbitrary real Banach spaces. Let ℒ be the Banach algebra of operators generated by theL-projections of a real Banach spaceU, and let ℳ (U ^* be the bounded operators on the dual spaceU ^* with adjoint in ℒ(U ^**. Then the adjoint operation maps ℒ (U) onto ℳ (U ^*). In particular, anyM-projection ofU ^* is weak^* continuous. Supported in part by the National Science Foundation.     

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.     

Trace formulae forS 1 invariant green’s operators onS 2

Summary We study the spectrum of the Laplacian acting on 1-forms for a surface of revolution diffeomorphic toS ² and obtain, for theS ¹ invariant spectrum, a trace formula for the eigenvalues of its Green’s operator. The trace formula is used to “hear” negative curvature for some metrics and to prove the existence of metrics onS ² with rather unusual spectral characteristics. This article was processed by the author using the L^AT_EX style file from Springer-Verlag.     

Compactly supported box-spline wavelets

A general procedure for constructing multivariate non-tensor-product wavelets that generate an orthogonal decomposition ofL ²(R)^s,s s≥1, is described and applied to yield explicit formulas for compactly supported spline-wavelets based on the multiresolution analysis ofL ²(R)^s 1≤s≤3, generated by any box spline whose direction set constitutes a unimodular matrix. In particular, when univariate cardinal B-splines are considered, the minimally supported cardinal spline-wavelets of Chui and Wang are recovered. A refined computational scheme for the orthogonalization of spaces with compactly supported wavelets is given. A recursive approximation scheme for “truncated” decomposition sequences is developed and a sharp error bound is included. A condition on the symmetry or anti-symmetry of the wavelets is applied to yield symmetric box-spline wavelets. Partially supported by ARO Grant DAAL 03-90-G-0091 Partially supported by NSF Grant DMS 89-0-01345 Partially supported by NATO Grant CRG 900158.     

TT-eigentensors for the Lichnerowicz Laplacian on some asymptotically hyperbolic manifolds with warped products metrics

Let (M =]0, ∞[×N, g) be an asymptotically hyperbolic manifold of dimension n + 1 ≥ 3, equipped with a warped product metric. We show that there exist no TT L ²-eigentensors with eigenvalue in the essential spectrum of the Lichnerowicz Laplacian Δ _L . If (M, g) is the real hyperbolic space, there is no symmetric L ²-eigentensors of Δ _L .     

Lattice-embedding scales ofL p spaces into orlicz spaces

We study the setP _X of scalarsp such thatL ^p is lattice-isomorphically embedded into a given rearrangement invariant (r.i.) function spaceX[0, 1]. Given 0<α≤β<∞, we construct a family of Orlicz function spacesX=L ^F [0, 1], with Boyd indicesα andβ, whose associated setsP _X are the closed intervals [γ, β], for everyγ withα≤γ≤β. In particular forα>2, this proves the existence of separable 2-convex r.i. function spaces on [0,1] containing isomorphically scales ofL ^p -spaces for different values ofp. We also show that, in general, the associated setP _X is not closed. Similar questions in the setting of Banach spaces with uncountable symmetric basis are also considered. Thus, we construct a family of Orlicz spaces ℓ ^F (I), with symmetric basis and indices fixed in advance, containing ℓ ^p (Γ-subspaces for differentp’s and uncountable Λ⊂I. In contrast with the behavior in the countable case (Lindenstrauss and Tzafriri [L-T₁]), we show that the set of scalarsp for which ℓ ^p (Γ) is isomorphic to a subspace of a given Orlicz space ℓ ^F (I) is not in general closed. Supported in part by DGICYT grant PB 94-0243.     

Subspaces ofL p (G) spanned by characters: 0<p<1

LetG be an infinite compact abelian group,μ a Borel measure onG with spectrumE, and 0<p<1. We show that ifμ is not absolutely continuous with respect to Haar measure, thenL _E ^P (G), the closure inL ^p (G) of theE-trigonometric polynomials, does not have enough continuous linear functionals to separate points. Ifμ is actually singular, thenL _E ^p (G) does not have any nontrivial continuous linear functionals at all. Our methods recover the classical F. and M. Riesz theorem, and a related several variable result of Bochner; they reveal the existence of small sets of characters that spanL ^P (T), where T is the unit circle; and they show that theH ^p spaces of the “big disc algebra” have one-dimensional dual.