Symmetric block bases in finite-dimensional normed spaces

It is shown that for 1 ≦p < ∞, any basisC-equivalent to the unit vector basis ofl _p ⁿ has a (1 + ε)-symmetric block basis of cardinality proportional ton/logn. When 1 <p < ∞, an upper bound proportional ton log logn/logn is also obtained. These results extend results of Amir and Milman in [2].     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

Averaging distances in real quasihypermetric Banach spaces of finite dimension

The average distance theorem of Gross implies that for each realN-dimensional Banach space (N≥2) there is a unique positive real numberr(E) with the following property: For each positive integern and for all (not necessarily distinct)x ₁,x ₂, …,x _n inE with ‖x ₁‖=‖x ₂‖=…=‖x _n‖=1, there exists anx inE with ‖x‖=1 such that The main result of this paper shows, thatr(E)≤2−1/N for each realN-dimensional Banach spaceE (N≥2) with the so-called quasihypermetric property (which is equivalent toE isL ¹-embeddable). Moreover, equality holds if and only ifE is isometrically isomorphic to ℝ ^N equipped with the usual 1-norm.     

Analogues of the “zero-two” law for positive linear contractions inL p andC(X)

LetT be a positive linear contraction inL ^p (1≦p<∞), then we show that lim ‖T ^pf −T ⁿ⁺¹ f‖ _p ≦(1 − ε)2^1/p (f∈L _p ⁺ , ε>0 independent off) implies already lim_n n→∞ ‖T ⁿf −T ⁿ⁺¹ n+1f ‖_p p=0. Several other related results as well as uniform variants of these are also given. Finally some similar results inLsu/t8 andC(X) are shown.     

On extremal point distributions in the Euclidean plane

We ask for the maximum σ _n ^γ of Σ _i,j=1 ⁿ ‖x _i-x _j‖^γ, where x ₁,χ,x _n are points in the Euclidean plane R ² with ‖x_i-x_j‖ ≦1 for all 1≦ i,j ≦ n and where ‖.‖^γ denotes the γ-th power of the Euclidean norm, γ ≧ 1. (For γ =1 this question was stated by L. Fejes Tóth in [1].) We calculate the exact value of σ _n ^γ for all γ γ 1,0758χ and give the distributions which attain the maximum σ _n ^γ . Moreover we prove upper bounds for σ _n ^γ for all γ ≧ 1 and calculate the exact value of σ ₄ ^γ for all γ ≧ 1. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Johnson–Lindenstrauss Lemma Almost Characterizes Hilbert Space, But Not Quite

Let X be a normed space that satisfies the Johnson–Lindenstrauss lemma (J–L lemma, in short) in the sense that for any integer n and any x ₁,…,x _n ∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that ‖x _i −x _j ‖≤‖L(x _i )−L(x _j )‖≤O(1)⋅‖x _i −x _j ‖ for all i,j∈{1,…,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion 2^2O(log*n)2^{2^{O(\log^{*}n)}} . On the other hand, we show that there exists a normed space Y which satisfies the J–L lemma, but for every n, there exists an n-dimensional subspace E _n ⊆Y whose Euclidean distortion is at least 2^Ω(α(n)), where α is the inverse Ackermann function.     

Graphs whose every transitive orientation contains almost every relation

Given a graphG onn vertices and a total ordering ≺ ofV(G), the transitive orientation ofG associated with ≺, denotedP(G; ≺), is the partial order onV(G) defined by settingx<y inP(G; ≺) if there is a pathx=x ₁ x ₂…x _r=y inG such thatx ₁ ≺x _j for 1≦i<j≦r. We investigate graphsG such that every transitive orientation ofG contains ₂ ⁿ −o(n ²) relations. We prove that almost everyG _n,p satisfies this requirement if , but almost noG _n,p satisfies the condition if (pn log log logn)/(logn log logn) is bounded. We also show that every graphG withn vertices and at mostcn logn edges has some transitive orientation with fewer than ₂ ⁿ −δ(c)n ² relations. Partially supported by MCS Grant 8104854.     

On some characterization of probability distributions in Hilbert spaces

Summary The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces:Theorem. — Let x₁, x₂, …, x_n be n (n≥3) independent random variables in the Hilbert spaceH, having their characteristic functionals f_k(t) = E[e^i(t,x _k)], (k=1, 2, …, n): let y₁=x₁ + x_n, y₂=x₂ + x_n, …, y_n−1=x_n−1 + x_n. If the characteristic functional f(t₁, t₂, …, t_n−1) of the random variables (y₁, y₂, …, y_n−1) does not vanish, then the joint distribution of (y₁, y₂, …, y_n−1) determines all the distributions of x₁, x₂, …, x_n up to change of location.     

Concerning the order of approximation of periodic continuous functions by trigonometric interpolation polynomials

Letx _kn=2θk/n,k=0,1 …n−1 (n odd positive integer). LetR _n(x) be the unique trigonometric polynomial of order 2n satisfying the interpolatory conditions:R _n(x_kn)=f(x_kn),R _n ^(j)(x_kn)=0,j=1,2,4,k=0,1…,n−1. We setw ₂(t,f) as the second modulus of continuity off(x). Then we prove that |R _n(x)-f(x)|=0(nw₂(1/nf)). We also examine the question of lower estimate of ‖R _n-f‖. This generalizes an earlier work of the author.     

Embedding ofl ∞ k in finite dimensional Banach spaces

Letx ₁,x ₂, ...,x _n ben unit vectors in a normed spaceX and defineM _n =Ave{‖Σ _i=1 ⁿ ε₁ x _i ‖:ε₁=±1}. We prove that there exists a setA⊂{1, ...,n} of cardinality such that {x _i } _i∈A is 16M _n -isomorphic to the natural basis ofl _∞ ^k . This result implies a significant improvement of the known results concerning embedding ofl _∞ ^k in finite dimensional Banach spaces. We also prove that for every ∈>0 there exists a constantC(∈) such that every normed spaceX _n of dimensionn either contains a (1+∈)-isomorphic copy ofl ₂ ^m for somem satisfying ln lnm≧1/2 ln lnn or contains a (1+∈)-isomorphic copy ofl _∞ ^k for somek satisfying ln lnk>1/2 ln lnn−C(∈). These results follow from some combinatorial properties of vectors with ±1 entries. The contribution of the first author to this paper forms part of his Ph.D. Thesis written under the supervision of Prof. M. A. Perles from the Hebrew University.     

The uniform approximation property in Orlicz spaces

It is proved that for every reflexive Orlicz spaceX there is a functionn(k,ε) so that wheneverE is ak-dimensional subspace ofX there exists an operatorT: X→X such thatT _1E=identity, ‖T‖≦1+ε and dimTX≦n(k,ε). Some general facts concerning the uniform approximation property are also presented. Research of the first named author was partially supported by NSF Grant MPS 74-07509-A01.     

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.     

Finitary reconstruction of a measure preserving transformation

This paper considers thefinitary reconstruction of an ergodic measure preserving transformationT of a complete separable metric spaceX from a single trajectoryx, Tx, …, or more generally, from a suitable reconstruction sequence x=x ₁,x ₂, … withx _i∈X. Ann-sample reconstruction is a functionT _n: X ⁿ⁺¹ →X; the map (·;x ₁, …,x _n)is treated as an estimate ofT(·) based on then initial elements of x. Given a reference probability measureμ ₀ and constantM>1, functionsT ₁,T ₂, … are defined, and it is shown that for everyμ with 1/M≤dμ/dμ ₀≤M, everyμ-preserving transformationT, and every reconstruction sequence x forT, the estimates (·;x ₁, …,x _nconverge toT in the weak topology. For the family of interval exchange transformations of [0, 1] a simple family of estimates is described and shown to be consistent both pointwise and in the strong topology. However, it is also shown that no finitary estimation scheme is consistent in the strong topology for the family of all ergodic Lebesgue measure preserving transformations of the unit interval, even if x is assumed to be a generic trajectory ofT. Supported in part by NSF Grant DMS-9501926.     

On extremal problems of graphs and generalized graphs

Anr-graph is a graph whose basic elements are its vertices and r-tuples. It is proved that to everyl andr there is anε(l, r) so that forn>n ₀ everyr-graph ofn vertices andn ^{r−ε(l, r)} r-tuples containsr. l verticesx ^(j), 1≦j≦r, 1≦i≦l, so that all ther-tuples occur in ther-graph.     

Divisibility properties of certain recurrent sequences

Let g and m be two positive integers, and let F be a polynomial with integer coefficients. We show that the recurrent sequence x₀ = g, x_n = x _n−1 ⁿ + F(n), n = 1, 2, 3,…, is periodic modulo m. Then a special case, with F(z) = 1 and with m = p > 2 being a prime number, is considered. We show, for instance, that the sequence x₀ = 2, x_n = x _n−1 ⁿ + 1, n = 1, 2, 3, …, has infinitely many elements divisible by every prime number p which is less than or equal to 211 except for three prime numbers p = 23, 47, 167 that do not divide x_n. These recurrent sequences are related to the construction of transcendental numbers ζ for which the sequences [ζ^n!], n = 1, 2, 3, …, have some nice divisibility properties. Bibliography: 18 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 76–82.     

The additive subgroup generated by a polynomial

SupposeR is a prime ring with the centerZ and the extended centroidC. Letp(x ₁, …,x _n) be a polynomial overC in noncommuting variablesx ₁, …,x _n. LetI be a nonzero ideal ofR andA be the additive subgroup ofRC generated by {p(a ₁, …,a _n):a ₁, …,a _n ∈I}. Then eitherp(x ₁, …,x _n) is central valued orA contains a noncentral Lie ideal ofR except in the only one case whereR is the ring of all 2 × 2 matrices over GF(2), the integers mod 2.     

Banded perturbations of the unit vector basis in some sequence spaces

Perturbations of the unit vector basis of the formX _n =Σ_|j−n|≦m a _nj e _j wherem is a fixed positive integer are investigated. It is shown that if |a _nj |≦1 and if {x _n } possesses a biorthogonal sequence uniformly bounded inl _p for some 1<=p<∞, then {x _n } is a seminormalized basic sequence in some reflexive Orlicz spacel _N, then {x_n} is equivalent to {e _n} inl _N.     

Eventually areally meanp-valent functions

Theorems concerning areally meanp-valent functions are extended to eventually areally meanp-valent functions. In particular, suppose is eventually areally meanp-valent in the unit disc,b, c are positive integers,a≧max {p−1, 0}. If |a _n|≦Cn ^α for alln=bm+c,m=1, 2, …, then |a _n|≦C′n ^α for alln. This is a marked extension of results due to Goluzin and to Hayman.     

The zero-two law in Banach lattice algebras

LetA be a unital Banach lattice algebra and leta εA ⁺ satisfy ‖a ‖≦1. Then either ‖a ⁿ⁺¹ −a ⁿ ‖=2 for alln≧0 or else ‖a ⁿ⁺¹ −a ⁿ ‖ → 0 asn → ∞. Cyclicity of the peripheral spectrum ofa is also established.     

Asymptotic behavior of nonexpansive mappings in normed linear spaces

LetT be a nonexpansive mapping on a normed linear spaceX. We show that there exists a linear functional.f, ‖f‖=1, such that, for allx∈X, lim_n→x f(T ⁿ x/n)=lim_n→x‖T ⁿ x/n ‖=α, where α≡inf _y∈c ‖Ty-y‖. This means, ifX is reflexive, that there is a faceF of the ball of radius α to whichT ⁿ x/n converges weakly for allx (inf_z∈f g(T ⁿ x/n-z)→0, for every linear functionalg); ifX is strictly conves as well as reflexive, the convergence is to a point; and ifX satisfies the stronger condition that its dual has Fréchet differentiable norm then the convergence is strong. Furthermore, we show that each of the foregoing conditions on X is satisfied if and only if the associated convergence property holds for all nonexpansiveT. Supported by National Science Foundation Grant MCS-79-066.