Unconditional Constant Research Articles

Unconditional convergence constants of g-frame expansions

In this paper, we prove that the unconditional constants of the g-frame expansion in a Hilbert space are bounded by [Formula: see text], where [Formula: see text], [Formula: see text] are the frame bounds of the g-frames. It follows that tight g-frames have unconditional constant one. Then we generalize this to a classification of such g-frames by showing that a g-Bessel sequence has unconditional constant one if it is an orthogonal sum of g-tight frames. We also obtain a new result under which a g-Bessel sequence is a g-frame from the view of unconditional constant. Finally, we prove similar results for cross g-frame expansions as long as the cross g-frame expansions stay uniformly bounded away from zero.

Monomial convergence for holomorphic functions on ℓr

Let $$\mathcal{F}$$ be either the set of all bounded holomorphic functions or the set of all m-homogeneous polynomials on the unit ball of lr. We give a systematic study of the sets of all u ∊ lr for which the monomial expansion $$\sum\nolimits_\alpha {{{{\partial ^\alpha }f(0)} \over {\alpha !}}{u^\alpha }} $$ of every f ∊ $$\mathcal{F}$$ converges. Inspired by recent results from the general theory of Dirichlet series, we establish as our main tool independently interesting, upper estimates for the unconditional basis constants of spaces of polynomials on lr spanned by finite sets of monomials.

Extreme 2-homogeneous polynomials on the plane with a hexagonal norm and applications to the polarization and unconditional constants

We classify the extreme 2-homogeneous polynomials on 2 with the hexagonal norm of weight ½. As applications, using its extreme points with the Krein-Milman Theorem, we explicitly compute the polarization and unconditional constants of .

The unconditional constants for Hilbert space frame expansions

The most fundamental notion in frame theory is the frame expansion of a vector. Although it is well known that these expansions are unconditionally convergent series, no characterizations of the unconditional constant were known. This has made it impossible to get accurate quantitative estimates for problems which require using subsequences of a frame. We will prove some new results in frame theory by showing that the unconditional constants of the frame expansion of a vector in a Hilbert space are bounded by BA, where A,B are the frame bounds of the frame. Tight frames thus have unconditional constant one, which we then generalize by showing that Bessel sequences have frame expansions with unconditional constant one if and only if the sequence is an orthogonal sum of tight frames. We give further results concerning frame expansions, in which we examine when BA is actually attained or not. We end by discussing the connections of this work to frame multipliers. These results hold in both real and complex Hilbert spaces.

The fixed point property for some generalized nonexpansive mappings and renormings

In 2008, T. Suzuki [33] defined one of most relevant extensions of the notion of nonexpansivity. This notion was extended in [15] defining the, so called, Cλ-mappings. In the last years, some papers have appeared trying to extend the most important results about existence of fixed points for nonexpansive mappings to this wider class of mappings (see, for instance, [2,5,9,10,15]). In this paper we continue this project proving that Banach spaces with extended unconditional bases satisfy the fixed point property for Cλ-mappings if the unconditional constant of the basis is small enough. We also begin a new project on renorming with the fixed point property for Cλ-mappings, proving that any separable Banach space can be renormed to satisfy the fixed point property for Cλ-mappings in a stable sense, that is, the space with the new norm satisfies the fixed point property which, in addition, is inherited by those isomorphic spaces which are close to it in the sense of the Banach–Mazur distance.

On the Bellman function of Nazarov, Treil and Volberg

We give an explicit formula for the Bellman function associated with the dual bound related to the unconditional constant of the Haar system.

Unconditional constants and polynomial inequalities

If P is a polynomial on R m of degree at most n , given by P ( x ) = ∑ α ∈ N m , | α | ≤ n a α x α , and P n ( R m ) is the space of such polynomials, then we define the polynomial | P | by | P | ( x ) = ∑ α ∈ N m , | α | ≤ n | a α | x α . Now if B ⊆ R m is a convex set, we define the norm ‖ P ‖ B ≔ sup { | P ( x ) | : x ∈ B } on P n ( R m ) , and then we investigate the inequality ⦀ P ⦀ B ≤ C B ‖ P ‖ B , providing sharp estimates on C B for some specific spaces of polynomials. These C B ’s happen to be the unconditional constants of the canonical bases of the considered spaces.

APPROXIMATIONS OF HOLOMORPHIC FUNCTIONS IN CERTAIN BANACH SPACES

Let E be a Banach space and let B(R)⊂E denote the open ball with centre at 0 and radius R. The following problem is studied: given 0<r<R, ∊>0 and a function f holomorphic on B(R), does there always exist an entire function g on E such that |f-g|<∊ on B(r)? L. Lempert has proved that the answer is positive for Banach spaces having an unconditional basis with unconditional constant 1. In this paper a somewhat shorter proof of Lemperts result is given. In general it is not possible to approximate f by polynomials since f does not need to be bounded on B(r).

Boundary Value Problems and Sharp Inequalities for Martingale Transforms

Let $p^\ast$ be the maximum of $p$ and $q$ where $1 0$. This improves an earlier inequality of the author by giving the best constant and conditions for equality. The inequality holds with the same constant if $\varepsilon$ is replaced by a real-valued predictable sequence uniformly bounded in absolute value by 1, thus yielding a similar inequality for stochastic integrals. The underlying method rests on finding an upper or a lower solution to a novel boundary value problem, a problem with no solution (the upper is not equal to the lower solution) except in the special case $p = 2$. The inequality above, in combination with the work of Ando, Dor, Maurey, Odell, Olevskii, Pelczynski, and Rosenthal, implies that the unconditional constant of a monotone basis of $L^p(0, 1)$ is $p^\ast - 1$. The paper also contains a number of other sharp inequalities for martingale transforms and stochastic integrals. Along with other applications, these provide answers to some questions that arise naturally in the study of the optimal control of martingales.

A nonlinear partial differential equation and the unconditional constant of the Haar system in 𝐿^{𝑝}

A nonlinear partial differential equation and the unconditional constant of the Haar system in 𝐿^{𝑝}

A note on the gl constant ofE $$\mathop \otimes \limits^ \vee $$ F F

LetX andY be Banach spaces. TFAE (1)X andY do not contain subspaces uniformly isomorphic to\(l_\infty ^n ,s.\) (2) The local unconditional structure constant of the space of bounded operatorsL (X*k,Y k) tends to infinity for every increasing sequence\(\left\{ {X_k } \right\}_{k = 1}^\infty \) and\(\left\{ {Y_k } \right\}_{k = 1}^\infty \) of finite-dimensional subspaces ofX andY respectively.

Large subspaces ofl ∞ n and estimates of the gordon-lewis constantand estimates of the gordon-lewis constant

Lower bounds are obtained for thegl constants and hence also for the unconditional basis constants of subspaces of finite dimensional Banach spaces. Sharp results are obtained for subspaces ofl ∞ n , while in the general case thegl constants of “random large” subspaces are related to the distance of “random large” subspaces to Euclidean spaces. In addition, a new isometric characterization ofl ∞ n is given, some new information is obtained concerningp-absolutely summing operators, and it is proved that every Banach space of dimensionn contains a subspace whose projection constant is of ordern 1/2.

On Basis Constants and Duality in Banach Spaces

AbstractEvery Banach space with a non-shrinking (unconditional) basis (Xi) can be renormed so that the biorthogonal sequence has a much smaller (unconditional) basis constant than (xi). On the other hand, if the unconditional constant of is C < 2 then the unconditional constant of (xi) is at most C/(2—C). This estimate is sharp.

The projection constant of finite-dimensional spaces whose unconditional basis constant is 1

The relations of the projection constant λ(E) and the isomorphic distance d(E, 1n∞) of finite-dimensional spacesE whose unconditional basis constant is 1 are investigated. It turns out that both are proportional to the norm of a certain vector inE.

A relation between diagonal and unconditional basis constants

This note establishes an inequality relating the weakly nuclear norm of the identity on a finite dimensional space with the diagonal symmetry constant of the space. An application shows the non-existance of"good" diagonally symmetric bases in L(l~), the space of linear operators on n-dimensional Hilbert space. More precisely, for any basis (bi)i _n t/2 It u 11/18. In addition it is shown that there are spaces with diagonal symmetry constant one, but with unconditional basis constant greater than one.