Extension of positive functionals on certain ordered spaces

An ordered linear spaceL is said to satisfy extension property (E1) if for every directed subspaceM ofL and positive linear functional ϕ onM, ϕ can be extended toL. A Riesz spaceL is said to satisfy extension property (E2) if for every sub-Riesz spaceM ofL and every real valued Riesz homomorphism ϕ onM, ϕ can be extended toL as a Riesz homomorphism. These properties were introduced by Schmidt in [5]. In this paper, it is shown that an ordered linear space has extension property (E1) if and only if it is order isomorphic to a function spaceL′ defined on a setX′ such that iff andg belong toL′ there exists a finite disjoint subsetM of the set of functions onX′ such that each off andg is a linear combination of the points ofM. An analogous theorem is derived for Riesz spaces with extension property (E2).     

Optimal recovery of linear functionals on sets of finite dimension

Suppose that X is a linear space and L ₁, …, L _n is a system of linearly independent functionals on P, where P ? X is a bounded set of dimension n + 1. Suppose that the linear functional L ₀ is defined in X. In this paper, we find an algorithm that recovers the functional L ₀ on the set P with the least error among all linear algorithms using the information L ₁ f, …, L _n f, f ∈ P.     

Certain finite dimensional maps and their application to hyperspaces

In [8] Y. Sternfeld and this author gave a positive answer to the following longstanding open problem: Is the hyperspace (=the space of all subcontinua endowed with the Hausdorff metric) of a 2-dimensional continuum infinite dimensional? This result was improved in [9] where it was shown that for every positive integer numbern a 2-dimensional continuum contains a 1-dimensional subcontiuum with hyperspace of dimension ≥n. And it was asked there: Does a 2-dimensional continuum contain a 1-dimensional subcontinuum with infinite dimensional hyperspace? In this note we answer this question in the positive. Our proof applies maps with the following properties. A real valued mapf on a compactumX is called a Bing map if every continuum that is contained in a fiber off is hereditarily indecomposable.f is called ann-dimensional Lelek map if the union of all non-trivial continua which are contained in the fibers off isn-dimensional. It is shown that for dimX=n+1 the maps which are both Bing andn-dimensional Lelek maps form a denseG _σ-subset of the function spaceC(X, I)     

Archimedean lattices

Anarchimedean lattice is a complete algebraic latticeL with the property that for each compact elementc∈L, the meet of all the maximal elements in the interval [0,c] is 0.L ishyper-archimedean if it is archimedean and for eachx∈L, [x, 1] is archimedean. The structure of these lattices is analysed from the point of view of their meet-irreducible elements. If the lattices are also Brouwer, then the existence of complements for the compact elements characterizes a particular class of hyper-archimedean lattices. The lattice ofl-ideals of an archimedean lattice ordered group is archimedean, and that of a hyper-archimedean lattice ordered group is hyper-archimedean. In the hyper-archimedean case those arising as lattices ofl-ideals are fully characterized.     

Generating sets for lattices of dimension two

The dimension of a partially ordered set P is the smallest integer n (if it exists) such that the partial order on P is the intersection of n linear orders. It is shown that if L is a lattice of dimension two containing a sublattice isomorphic to the modular lattice M_2n+1, then every generating set of L has at least n+2 elements. A consequence is that every finitely generated lattice of dimension two and with no infinite chains is finite.     

On the dimension of ordered sets with the 2-cutset property

An ordered setP is said to have 2-cutset property if, for every elementx ofP, there is a setS of elements ofP which are noncomparable tox, with |S|?2, such that every maximal chain inP meets {x}∪S. We consider the following question: Does there exist ordered sets with the 2-cutset property which have arbitrarily large dimension? We answer the question in the negative by establishing the following two results.Theorem: There are positive integersc andd such that every ordered setP with the 2-cutset property can be represented asP=X∪Y, whereX is an ordinal sum of intervals ofP having dimension ?d, andY is a subset ofP having width ?c. Corollary: There is a positive integern such that every ordered set with the 2-cutset property has dimension ?n.     

A note on reflexivity in Banach spaces

For any normed spaceX, the unit ball ofX is weak *-dense in the unit ball ofX ^**. This says that for any ε>0, for anyF in the unit ball ofX ^**, and for anyf ₁,…,f _n inX ^*, the system of inequalities |f _i(x)?F(f _i)|≤ε can be solved for somex in the unit ball ofX. The author shows that the requirement that ε be strictly positive can be dropped only ifX is reflexive.     

A subclass of Ockham algebras

Here we introduce a subclass of the class of Ockham algebras ( L ; f ) for which L satisfies the property that for every x ∈ L , there exists n ≥ 0 such that fn ( x ) and fn+1 ( x ) are complementary. We characterize the structure of the lattice of congruences on such an algebra ( L ; f ). We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.     

A note on shadowing with chain transitivity

Let f : X → X be a continuous map of a compact metric space X. The map f induces in a natural way a map f_M on the space M(X) of probability measures on X, and a transformation f_K on the space K(X) of closed subsets of X. In this paper, we show that if (X, f) is a chain transitive system with shadowing property, then exactly one of the following two statements holds:

(a)

fⁿ and (f_K)ⁿ are syndetically sensitive for all n ? 1.

(b)

fⁿ and (f_K)ⁿ are equicontinuous for all n ? 1.

In particular, we show that for a continuous map f : X → X of a compact metric space X with infinite elements, if f is a chain transitive map with the shadowing property, then fⁿ and (f_K)ⁿ are syndetically sensitive for all n ? 1. Also, we show that if f_M (resp. f_K) is chain transitive and syndetically sensitive, and f_M (resp. f_K) has the shadowing property, then f is sensitive.In addition, we introduce the notion of ergodical sensitivity and present a sufficient condition for a chain transitive system (X, f) (resp. (M(X), f_M)) to be ergodically sensitive. As an application, we show that for a L-hyperbolic homeomorphism f of a compact metric space X, if f has the AASP, then fⁿ is syndetically sensitive and multi-sensitive for all n ? 1.     

Algebraic form of a theorem of Hahn—Banach type for lattice manifolds

Let E be a vector lattice. A linear functionalf on E is called a lattice homomorphism iff(sup (x, y)) = max (f(x),f(y)) for all x, y E. For lattice homomorphisms a theorem of Hahn—Banach type is valid. In this note we prove an algebraic analog of this theorem.Translated from Matematicheskie Zametki, Vol. 16, No. 4, pp. 595–600, October, 1974.In conclusion the author expresses his thanks to D. A. Raikov for his statement of the problem and his interest in my work.     

Constrained best approximation in Hilbert space III. Applications ton-convex functions

This paper continues the study of best approximation in a Hilbert spaceX from a subsetK which is the intersection of a closed convex coneC and a closed linear variety, with special emphasis on application to then-convex functions. A subtle separation theorem is utilized to significantly extend the results in [4] and to obtain new results even for the “classical” cone of nonnegative functions. It was shown in [4] that finding best approximations inK to anyf inX can be reduced to the (generally much simpler) problem of finding best approximations to a certain perturbation off from either the coneC or a certain subconeC _F. We will show how to determine this subconeC _F, give the precise condition characterizing whenC _F=C, and apply and strengthen these general results in the practically important case whenC is the cone ofn-convex functions inL ₂ (a,b),     

On then-cutset property

For a partially ordered setP and an elementx ofP, a subsetS ofP is called a cutset forx inP if every element ofS is noncomparable tox and every maximal chain ofP meets {x}∪S. We letc(P) denote the smallest integerk such that every elementx ofP has a cutsetS with ‖S‖?k: Ifc(P)?n we say thatP has then-cutset property. Our results bear on the following question: givenP, what is the smallestn such thatP can be embedded in a partially ordered set having then-cutset property? As usual, 2 ⁿ denotes the Boolean lattice of all subsets of ann-element set, andB _n denotes the set of atoms and co-atoms of 2 ⁿ . We establish the following results: (i) a characterization, by means of forbidden configurations, of whichP can be embedded in a partially ordered set having the 1-cutset property; (ii) ifP contains a copy of 2 ⁿ , thenc(P)?2^[n/2]?1; (iii) for everyn>3 there is a partially ordered setP containing 2 ⁿ such thatc(P)<c(2 ⁿ ); (iv) for every positive integern there is a positive integerN such that, ifB _m is contained in a partially ordered set having then-cutset property, thenm?N.     

A criterion for the uniform distribution of sequences in compact metric spaces

LetX be a compact metric space, le μ be a non-negative normalized Borel measure onX and letf be a measurable bounded real-valued function defined onX such thatf is μ-almost everywhere continuous and different from zero. It is proved that a sequence (x _n ),n=1,2, … of points inX is μ-uniformly distributed if and only if for every Borel setE?X with μ(Bd(E))=0 we have \(\mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{n = 1}^N {f(x_n )} 1_E (x_n ) = \int\limits_E {f(x)d\mu (x)} ,\) where 1 _E denotes the characteristic function ofE andbdE the boundary ofE. Furthermore some quantitative aspects and generalizations of this theorem are discussed.     

L p -spectral estimation with anL ∞-upper bound

Given the constraint 0≤f≤B, whereB is in the interior of the positive cone ofL _∞, and given a finite number of correlations off, we wish to estimatef. Since only a finite number of correlations are given, this does not uniquely determinef. We estimatef by picking the unique function Φ₀ satisfying the constraints and minimizing theL _p -norm with 1<p<∞. Under suitable conditions, the form of the solution is shown to be $$\Phi _0 (f) = \min \{ B(x), \max \{ 0,P(x)\} ^{1/(p - 1)} \} ,$$ whereP is a linear combination of the correlation functions.     

Best methods of interpolation for certain classes of differentiable functions

For the class W^(r)L_q (M;a, b), 1≤q≤∞, we construct the best method of approximation of the functionalf (x), x∈ [a, b], among all the methods using only information about the values off (^k)(x_i) (k=0, 1, ..., r?1; i=1, 2, ..., N).     

On symmetric basic sequences in lorentz sequence spaces II

It is shown that if {y _n} is a block of type I of a symmetric basis {x _n} in a Banach spaceX, then {y _n} is equivalent to {x _n} if and only if the closed linear span [y _n] of {y _n} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {x _n,f _n} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [f _n] has a complemented subspace isomorphic tol _p (respectively,l _q, 1/p+1/q=1 when 1<p<+∞ andc ₀ whenp=1) and numerous other results on complemented subspaces ofd(a, p) and [f _n] are obtained. We also obtain necessary and sufficient conditions such that [f _n] have exactly two non-equivalent symmetric basic sequences. Finally, we exhibit a Banach spaceX with symmetric basis {x _n} such that every symmetric block basic sequence of {x _n} spans a complemented subspace inX butX is not isomorphic to eitherc ₀ orl _p, 1≤p<+∞.     

An extension theorem for holomorphic functions of slow growth on covering spaces of projective manifolds

LetX be a projective manifold of dimension n ≥ 2 andY →X an infinite covering space. EmbedX into projective space by sections of a sufficiently ample line bundle. We prove that any holomorphic function of sufficiently slow growth on the preimage of a transverse intersection ofX by a linear subspace of codimension <n extends toY. The proof uses a Hausdorff duality theorem for L₂ cohomology. We also show that every projective manifold has a finite branched covering whose universal covering space is Stein.     

A Komlós Theorem for abstract Banach lattices of measurable functions

Consider a Banach function space X(μ) of (classes of) locally integrable functions over a σ-finite measure space (Ω,Σ,μ) with the weak σ-Fatou property. Day and Lennard (2010) [9] proved that the theorem of Komlós on convergence of Cesàro sums in L¹[0,1] holds also in these spaces; i.e. for every bounded sequence n(fn) in X(μ), there exists a subsequence k(fnk) and a function f∈X(μ) such that for any further subsequence j(hj) of k(fnk), the series converges μ-a.e. to f. In this paper we generalize this result to a more general class of Banach spaces of classes of measurable functions — spaces L¹(ν) of integrable functions with respect to a vector measure ν on a δ-ring — and explore to which point the Fatou property and the Komlós property are equivalent. In particular we prove that this always holds for ideals of spaces L¹(ν) with the weak σ-Fatou property, and provide an example of a Banach lattice of measurable functions that is Fatou but do not satisfy the Komlós Theorem.     

On linear extension majority graphs of partial orders

The linear extension majority (LEM) graph (X, > p) of a finite partially ordered set (X, P) has x>py for elements x and y in X just when more linear extensions L of P on X have xLy than yLx. A linear extension L of P on X is a linear order on X with P ? L. There exist finite partially ordered sets (X, P) whose LEM graphs have no >p-maximal elements, in which case every x in X has an x′ in X for which x′>px.