Two-dimensional real symmetric spaces with maximal projection constant

Chalmers, Bruce L.; Lewicki, Grzegorz

doi:10.4064/ap-73-2-119-134

Cited by 16 publications

(33 citation statements)

References 6 publications

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“…It is known [1] that the space G 3 , whose unit ball is a regular hexagon, maximizes the Banach-Mazur distance κ 2 ∞ (V) from V to 2 ∞ over all two-dimensional normed spaces V. It was also asserted [9] that it maximizes the projection constant p(V), but Chalmers [2] claimed that this original proof is incorrect. It is nonetheless natural to wonder if the hexagonal space maximizes all the intermediate quantities, i.e.…”

Section: Extremality Of the Hexagonal Spacementioning

confidence: 99%

Allometry constants of finite-dimensional spaces: theory and computations

Foucart

2009

Numer. Math.

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We describe the computations of some intrinsic constants associated to an n-dimensional normed space V, namely the N -th "allometry" constantsThese are related to Banach-Mazur distances and to several types of projection constants. We also present the results of our computations for some low-dimensional spaces such as sequence spaces, polynomial spaces, and polygonal spaces. An eye is kept on the optimal operators T and T , or equivalently, in the case N = n, on the best conditioned bases. In particular, we uncover that the best conditioned bases of quadratic polynomials are not symmetric, and that the Lagrange bases at equidistant nodes are best conditioned in the spaces of trigonometric polynomials of degree at most one and two.

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Section: Extremality Of the Hexagonal Spacementioning

confidence: 99%

Allometry constants of finite-dimensional spaces: theory and computations

Foucart

2009

Numer. Math.

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“…Let Q n APðH n ; H nÀ1 Þ be a minimal projection. Then critðQ n Þ (see (3)) consists of at least n points.…”

Section: Article In Pressmentioning

confidence: 99%

“…It is worth noting that there exists a large number of papers concerning minimal projections. Mainly the problems of existence [12,15], uniqueness [11,13,25,32,33], characterization of onecomplemented subspaces [1,2,26,30,31], concrete formulas for minimal projections [3][4][5][6][7]10,12,21,22,24,29,34], estimates of the relative projection constants [5,14,19,23,28,31,35], construction of spaces with large relative projection constants [4,5,[16][17][18][19][20], as well as the problems connected with shape-preserving projections [8,9] were considered. For basic information concerning this topic the reader is referred to [27].…”

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confidence: 99%

Codimension-one minimal projections onto Haar subspaces

Lewicki

Prophet

2004

Journal of Approximation Theory

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Let H n be an n-dimensional Haar subspace of X ¼ C R ½a; b and let H nÀ1 be a Haar subspace of H n of dimension n À 1: In this note we show (Theorem 6) that if the norm of a minimal projection from H n onto H nÀ1 is greater than 1, then this projection is an interpolating projection. This is a surprising result in comparison with Cheney and Morris () which shows that there is no interpolating minimal projection from C½a; b onto the space of polynomials of degree pn; ðnX2Þ: Moreover, this minimal projection is unique (Theorem 9). In particular, Theorem 6 holds for polynomial spaces, generalizing a result of Prophet [(J.

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“…The Chalmers-Metcalf theorem has many applications especially in the case of X = L 1 , e.g., [3][4][5][6][7]13,[21][22][23]35,38,39]. The aim of this paper is to relate some properties of this operator to the uniqueness of minimal projections.…”

Section: Introductionmentioning

confidence: 99%

Chalmers–Metcalf operator and uniqueness of minimal projections

Lewicki

Skrzypek

2007

Journal of Approximation Theory

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We know that not all minimal projections in L p (1 < p < ∞) are unique (see [B. Shekhtman, L. Skrzypek, On the non-uniqueness of minimal projections in L p spaces]). The aim of this paper is examine the connection of the Chalmers-Metcalf operator (introduced in [B.L. Chalmers, F.T. Metcalf, A characterization and equations for minimal projections and extensions, J. Oper. Theory 32 (1994) 31-46]) to the uniqueness of minimal projections. The main theorem of this paper is Theorem 2.2. It relates uniqueness of minimal projections to the invertibility of the Chalmers-Metcalf operator. It is worth mentioning that to a given minimal projection (even unique) we may find many different Chalmers-Metcalf operators, some of them invertible, some not-see Example 2.6. The main application is in Section 3, where we have proven that minimal projections onto symmetric subspaces in smooth Banach spaces are unique (Theorem 3.2). This leads (in Section 4) to the solution of the problem of uniqueness of classical Rademacher projections in L p [0, 1] for 1 < p < ∞.

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Two-dimensional real symmetric spaces with maximal projection constant

Cited by 16 publications

References 6 publications

Allometry constants of finite-dimensional spaces: theory and computations

Allometry constants of finite-dimensional spaces: theory and computations

Codimension-one minimal projections onto Haar subspaces

Chalmers–Metcalf operator and uniqueness of minimal projections

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