A characterization and equations for minimal shape-preserving projections

Chalmers, Bruce L.; Mupasiri, Douglas; Prophet, M. P.

doi:10.1016/j.jat.2005.11.006

Cited by 6 publications

(4 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We begin with a corollary given in [9]; it describes how the functionals that define a projection must be chosen in order for the projection to preserve shape.…”

Section: Proofs Of Minimalitymentioning

confidence: 99%

“…In fact, by Corollary 5.1 every element of P S σ (X, Π n ) can be constructed in this way. And that is why we are unable to appeal to the standard theory of minimal projections (described for example in [9]) which relies on best approximations from a linear space (and not from a cone). Therefore we proceed in the following way: we show that replacing δ n−1 1 −δ n−1 0 in P 0,n with any other allowable functional from S * results in an element of P S σ (X, Π n ) with norm at least as large as P 0,n .…”

Section: Proofs Of Minimalitymentioning

confidence: 99%

See 1 more Smart Citation

Minimal multi-convex projections

Lewicki

Prophet

2007

Studia Math.

View full text Add to dashboard Cite

Abstract. We say that a function from X = C L [0, 1] is k-convex (for k ≤ L) if its kth derivative is nonnegative. Let P denote a projection from X onto V = Π n ⊂ X, where Π n denotes the space of algebraic polynomials of degree less than or equal to n. If we want P to leave invariant the cone of k-convex functions (k ≤ n), we find that such a demand is impossible to fulfill for nearly every k. Indeed, only for k = n−1 and k = n does such a projection exist. So let us consider instead a more general "shape" to preserve. Let σ = (σ 0 , σ 1 , . . . , σ n ) be an (n + 1)-tuple with σ i ∈ {0, 1}; we say f ∈ X is multi-convex if f (i) ≥ 0 for i such that σ i = 1. We characterize those σ for which there exists a projection onto V preserving the multi-convex shape. For those shapes able to be preserved via a projection, we construct (in all but one case) a minimal norm multi-convex preserving projection. Out of necessity, we include some results concerning the geometrical structure of C L [0, 1].1. Introduction. When X is a Banach space and V ⊂ X a subspace, we denote by P(X, V ) the set of all projections from X onto V ; in the cases where there is no ambiguity, we will simply write P. We say that a projection P 0 is minimal if P 0 ≤ P for all P ∈ P(X, V ).There exist a large number of papers concerning minimal projections. The problems considered are mainly existence ([15] While a minimal projection will, in general, provide good approximations, it may fail to preserve particular properties of elements, as illustrated below. We are therefore motivated to look for projections which leave invari-

show abstract

“…We begin with a corollary given in [9]; it describes how the functionals that define a projection must be chosen in order for the projection to preserve shape.…”

Section: Proofs Of Minimalitymentioning

confidence: 99%

Section: Proofs Of Minimalitymentioning

confidence: 99%

Minimal multi-convex projections

Lewicki

Prophet

2007

Studia Math.

View full text Add to dashboard Cite

show abstract

“…One can appropriately modify the proof given in [5], Theorem 1, as follows. The problem is equivalent to best approximating, in the numerical-radius norm, a fixed operator T 0 ∈ T from the space of operators D = {∆ ∈ B :…”

Section: Theorem 21 (Characterization) T Is a Minimal Numerical-radmentioning

confidence: 99%

Minimal numerical-radius extensions of operators

Aksoy

Chalmers

2007

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. In this paper we characterize minimal numerical-radius extensions of operators from finite-dimensional subspaces and compare them with minimal operator-norm extensions. We note that in the cases L p , p = 1, ∞, and in the case of self-adjoint extensions in L 2 , the two extensions and their norms are equal.We also show that, in the case of L p , 1 < p < ∞, and more generally in the case of the dual space being strictly convex, if the minimal projections with respect to the operator norm and with respect to the numerical radius have equal norms, then the operator norm is 1. An analogous result is also true for an arbitrary extension. Finally, we provide an example of a projection from l p 3 onto a two-dimensional subspace which is minimal with respect to norm but not with respect to the numerical radius for p = 1, 2, ∞, and we determine the minimal numerical-radius projection in this same situation.

show abstract

“…The paper [5] gives a characterization of P S = ∅ under so-called high-dimensional assumptions (which are explained below). As illustrated, for example, in [1,4] and [6], there are many natural settings for which the high-dimensional assumptions are valid (and thus the characterization can be applied).…”

Section: Introductionmentioning

confidence: 99%

Shape-preserving projections in low-dimensional settings and the q-monotone case

Prophet

Shevchuk

2012

Ukr Math J

View full text Add to dashboard Cite

In this setting, one can ask if S is left invariant under P , i.e., if P S ⊂ S. If V is finite-dimensional and S is a cone with particular structure, then the occurrence of the imbedding P S ⊂ S can be characterized through a geometric description. This characterization relies heavily on the structure of S, or, more specifically, on the structure of the cone S * dual to S. In this paper, шe remove the structural assumptions on S * and characterize the cases where P S ⊂ S. We note that the (so-called) q-monotone shape forms a cone which (lacks structure and thus) serves as an application for our characterization. Нехай P : X → V-проекцiя дiйсного банахового простору X на пiдпростiр V i, крiм того, S ⊂ X. У цiй постановцi виникає питання: чи є S лiвоiнварiантним пiд дiєю P , тобто чи має мiсце вкладення P S ⊂ S? Якщо пiдпростiр V є скiнченновимiрним, а S є конусом iз певною структурою, то вкладення P S ⊂ S може бути охарактеризовано шляхом геометричного опису. Ця характеризацiя iстотно залежить вiд структури S, або, точнiше, вiд структури конуса S * , спряженого до S. У цiй роботi усунено структурнi припущення щодо S * i охарактеризовано випадки, у яких P S ⊂ S. Вiдзначено, що (так звана) q-монотонна форма утворює конус, який (не має структури i тому) може бути використаний для застосування нашої характеризацiї.

show abstract

A characterization and equations for minimal shape-preserving projections

Cited by 6 publications

References 10 publications

Minimal multi-convex projections

Minimal multi-convex projections

Minimal numerical-radius extensions of operators

Shape-preserving projections in low-dimensional settings and the q-monotone case

Contact Info

Product

Resources

About