On Orlicz sequence spaces. II

Given a separable Orlicz sequence spacel _F we investigate those Orlicz sequence spacesl _f which are isomorphic to subspaces (respectively complemented subspaces) ofl _F. We give in particular an example of a reflexive Orlicz sequence space which does not contain anyl _p, 1<p<∞, as a complemented subspace.     

Lattice-universal Orlicz spaces on probability spaces

Lattice-universal Orlicz function spacesL ^F _α,β[0, 1] with prefixed Boyd indices are constructed. Namely, given 0<α<β<∞ arbitrary there exists Orlicz function spacesL ^F _α,β[0, 1] with indices α and β such that every Orlicz function spaceL ^G [0, 1] with indices between α and β is lattice-isomorphic to a sublattice ofL ^F _α,β[0, 1]. The existence of classes of universal Orlicz spacesl ^Fα,β(I) with uncountable symmetric basis and prefixed indices α and β is also proved in the uncountable discrete case. Partially supported by BFM2001-1284.     

Onl p-complemented copies in Orlicz spaces

Let 1<α≦β<∞ andF be an arbitrary closed subset of the interval [α,β]. An Orlicz sequence spacel ^φ (resp. an Orlicz function spaceL ^φ(μ)) with associated indices α and β is found in such a way that the set of valuesp for which thel ^p-space is isomorphic to a complemented subspace ofl ^φ (resp.L ^φ(μ)) is precisely the given setF (resp.F ∪ {2}). Also, a recent result of Hernández and Peirats [1] is extended showing that, even for the case in which the indices satisfy α_φ ^∞<2<β_φ ^∞, there exist minimal Orlicz function spacesL ^φ(μ) with no complemented copy ofl ^p for anyp ≠ 2. Supported in part by CAICYT grant 0338-84.     

Two problems of Valdivia on distinguished Frechet spaces

In this paper we construct the following spaces: (a) a Fréchet spaceE with basis which is non-distinguished and has no subspace isomorphic to ℓ₁ (b) a non-distinguished Fréchet spaceF such that every separable subspace ofF is distinguished (even more, every separable subspace ofF has a separable dual). These examples answer in the negative two questions posed by Valdivia.     

Finite representability ofl p(X) in Orlicz function spaces

We show that ifl _p(X),p ≠ 2, is finitely crudely representable in an Orlicz spaceL _ϕ (which does not containc ₀) then the Banach spaceX is isomorphic to a subspace ofL _p. The same remains true forp = 2 whenL _ϕ is 2-concave or 2-convex, or ifX has local unconditional structure. We extend a theorem of Guerre and Levy to Orlicz function spaces.     

Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multiparameter case

In a sequence ofn independent random variables the pdf changes fromf(x, 0) tof(x, 0 + δv_n⁻¹) after the firstnλ variables. The problem is to estimateλ (0, 1 ), where 0 and δ are unknownd-dim parameters andv_n → ∞ slower thann^1/2. Let_n denote the maximum likelihood estimator (mle) ofλ. Analyzing the local behavior of the likelihood function near the true parameter values it is shown under regularity conditions that ifn_n²(− λ) is bounded in probability asn → ∞, then it converges in law to the timeT_(δjδ)^1/2 at which a two-sided Brownian motion (B.M.) with drift1/2(δ′Jδ)^1/2ton(−∞, ∞) attains its a.s. unique minimum, whereJ denotes the Fisher-information matrix. This generalizes the result for small change in mean of univariate normal random variables obtained by Bhattacharya and Brockwell (1976,Z. Warsch. Verw. Gebiete37, 51–75) who also derived the distribution ofT_μ forμ > 0. For the general case an alternative estimator is constructed by a three-step procedure which is shown to have the above asymptotic distribution. In the important case of multiparameter exponential families, the construction of this estimator is considerably simplified.     

Twisted sums and a problem of Klee

LetF be a quasi-linear map on a separable normed spaceE, and assume thatF splits on an infinite-dimensional subspace ofE. Then the twisted sum topology on ℝ⊗_F E can be written as the supremum of a nearly convex topology and a trivial dual topology. (This partially answers a question of Klee.) The result applies to the Ribe space and to James’s space. To victor Klee     

Utility functions and the ‘lin’ operation for convex sets

For an ℵ₀-dimensional spaceE of alternatives, there is described a preference relation ≿ such that (in a very strong sense)no information about ≿ can be expressed in terms of finite-dimensional linear transformations ofE. The same construction shows that for each countable ordinalβ, E contains a convex coneK such that lin ^β K =E but lin^β K ≡E forα<β.     

W(R)-splines

In [3] Golomb describes, for 1 < p < ∞, the H^r,p(R)-extremal extension F_* of a function ƒ:E → R (i.e., the H^r,p-spline with knots in E) and studies the cone H_*E^r,p of all such splines. We study the problem of determining when F_* is in W^r,p ≡ H^r,p ∩ L^p. If F_* ε W^r,p, then F_* is called a W^r,p-spline, and we denote by W_*E^r,p the cone of all such splines. If E is quasiuniform, then F_* ε W^r,p if and only if {ƒ(t_i)}_tiεE ε l^p. The cone W_*E^r,p with E quasiuniform is shown to be homeomorphic to l^p. Similarly, H_*E^r,p is homeomorphic to h^r,p. Approximation properties of the W^r,p-splines are studied and error bounds in terms of the mesh size ¦ E ¦ are calculated. Restricting ourselves to the case p = 2 and to quasiuniform partitions E, the second integral relation is proved and better error bounds in terms of ¦ E ¦ are derived.     

On orlicz sequence spaces III

It is proved that the set ofp's such thatl _p is isomorphic to a subspace of a given Orlicz spacel _Fforms an interval. Some examples and properties of minimal Orlicz sequence spaces are presented. It is proved that an Orlicz function space (different froml ₂) is not isomorphic to a subspace of an Orlicz sequence space. Finally it is shown (under a certain restriction) that if two Orlicz function spaces are isomorphic, then they are identical (i.e. consist of the same functions).     

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.     

The Banach-Mazur distance between symmetric spaces

We show that the Banach-Mazur distance betweenN-dimensional symmetric spacesE andF satisfies , wherec is a numerical constant. IfE is a symmetric space, then max (M ⁽²⁾(E),M ₍₂₎(E)), whereM ⁽²⁾(E) (resp.M ₍₂₎(E)) denotes the 2-convexity (resp. the 2-concavity) constant ofE. We also give an example of a spaceF with an 1-unconditional basis and enough symmetries that satisfiesd(F, l ₂ ^dimF )=M ⁽²⁾(F)M ₍₂₎(F). Partially supported by NSF Grant MCS-8201044.     

Embeddings ofC(Δ) andL 1[0, 1] in Banach lattices

It is proved that ifE is a separable Banach lattice withE′ weakly sequentially complete,F is a Banach space andT:E→F is a bounded linear operator withT′F′ non-separable, then there is a subspaceG ofE, isomorphic toC(Δ), such thatT _G is an isomorphism, whereC(Δ) denotes the space of continuous real valued functions on the Cantor discontinuum. This generalizes an earlier result of the second-named author. A number of conditions are proved equivalent for a Banach latticeE to contain a subspace order isomorphic toC(Δ). Among them are the following:L ¹ is lattice isomorphic to a sublattice ofE′;C(Δ)′ is lattice isomorphic to a sublattice ofE′; E contains an order bounded sequence with no weak Cauchy subsequence;E has a separable closed sublatticeF such thatF′ does not have a weak order unit. The research of both authors was partially supported by the National Science Foundation, NSF Grant No MPS 71-02839 A04.     

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach spaceℓ _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

An estimate of the error for eigen functions in the eigenfunction problem for the operator of the linear elasticity theory

We consider an eigenfunction problem for a system of Lamé equations in a three-dimensional parallelepiped in the case of a mixed boundary-value condition on the boundary. By using Steklov averaging operators, the approximation error is given in divergent form. The accuracy of the difference scheme is studied for generalized solutions from the spaceW ₂ ³(Ω). An O(|h|^1.5)-estimate is obtained for eigenfunctions in the grid metric ofW ₂ ¹(ω). Bibliography: 5 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 100–109.     

Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions

Forn2, let (μ^x_τ, n)_τ0be the distributions of the Brownian motion on the unit sphereS^{n n+1}starting in some pointxSⁿ. This paper supplements results of Saloff-Coste concerning the rate of convergence ofμ^x_τ, nto the uniform distributionU_nonSⁿforτ→∞ depending on the dimensionn. We show that,[formula]forτ_n:=(ln n+2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measuresμ^x_τ, nby measures which are known explicitly via Poisson kernels onSⁿ, and which tend, after suitable projections and dilatations, to normal distributions on forn→∞. The above result as well as some further related limit results will be derived in this paper in the slightly more general context of Jacobi-type hypergroups.     

On some classes of algebraic surfaces with an infinite set of subspaces of skew symmetry

Special classes of (m – 1)-dimensional algebraic surfaces F ⁿ in a space E^m with inifinite set ofsubspaces of skew symmetry (in particular, orthogonal) are studied. It is assumed that directions of symmetry, as a rule, are asymptotic for F _n .Translated from Dinamicheskie Sistemy, No. 8, pp. 119–126, 1989.     

Calderón--Lozanovskii; construction for mixed norm spaces

We show that the Calderón--Lozanovskii; construction φ(.) commutes with arbitrary mixed norm spaces, that is, φ(E₀[F₀], E₁[F₁]) = φ(E₀, E₁) [φ(F₀, F₁)] if and only if φ is equivalent to a power function. This result we obtain by giving characterizations of the corresponding embeddings of φ(E₀[F₀], E₁[F₁]) into φ₀ (E₀, E₁)[φ₁ (F₀, F₁)] and vice versa in terms of the functions φ, φ₀, φ₁. As a particular case, we get embeddings of an Orlicz space with mixed norms into an Orlicz space on a product of measure spaces. Applications to classical operators between mixed norm Orlicz spaces are also discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Nonparametric Approach for Non-Gaussian Vector Stationary Processes

Suppose that {z(t)} is a non-Gaussian vector stationary process with spectral density matrixf(λ). In this paper we consider the testing problemH: ∫^π_−π K{f(λ)} dλ=cagainstA: ∫^π_−π K{f(λ)} dλ≠c, whereK{·} is an appropriate function andcis a given constant. For this problem we propose a testT_nbased on ∫^π_−π K{f(λ)} dλ=c, wheref(λ) is a nonparametric spectral estimator off(λ), and we define an efficacy ofT_nunder a sequence of nonparametric contiguous alternatives. The efficacy usually depnds on the fourth-order cumulant spectraf₄^Zofz(t). If it does not depend onf₄^Z, we say thatT_nis non-Gaussian robust. We will give sufficient conditions forT_nto be non-Gaussian robust. Since our test setting is very wide we can apply the result to many problems in time series. We discuss interrelation analysis of the components of {z(t)} and eigenvalue analysis off(λ). The essential point of our approach is that we do not assume the parametric form off(λ). Also some numerical studies are given and they confirm the theoretical results.     

Sur les espaces de Banach contenantl 1(τ)

Letτ be a cardinal with cf(τ)>ℵ₀. Then a Banach spaceE contains a subspace isomorphic tol ^l(τ) if and only if [0,1] ^r is a continuous image of the unit ballE′₁ ofE′, provided with the w*-topology. It follows that, for each cardinalκ, ifE′₁ contains a copy ofβκ, thenE has a quotient isomorphic tol ^∞(κ). In this situation we show thatE has even a quotientisometric tol ^∞(κ).