Isometric embed-dings between classical Banach spaces, cubature formulas, and spherical designs

Lyubich, Yuri I.; Vaserstein, L. N.

doi:10.1007/bf01263664

Cited by 50 publications

(45 citation statements)

References 34 publications

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“…It turns out that the existence of such embeddings is a special phenomenon. In the real case the theorem was proved in [22] under the assumption q = 2 and later in [25] without this assumption. In §2 we present a proof over any K.…”

Section: §1 Introductionmentioning

confidence: 84%

“…Another proof given in [25] can be extended to the complex case; see [23]. This is impossible to do in the quaternion case for the reason mentioned above.…”

Section: In Other Words Any Function φ ∈ φ K (M P) Is a Linear Combmentioning

confidence: 99%

“…In [29] and [25] (two independent works, both related to K = R) the following important interpretation of (1.9) was observed: this is a cubature formula In such a way a rich collection of isometric embeddings over R was produced in [25,29].…”

Section: Theorem 1 If There Exists An Isometric Embeddingmentioning

confidence: 99%

“…Here n = p/2 + 1 because the centrally symmetric nodes can be joined. Moreover, this value of n turns out to be minimal [25,29].…”

Section: Theorem 1 If There Exists An Isometric Embeddingmentioning

confidence: 99%

“…needs a small modification of that argument; see [25,29]. (Note that (1.12) for K = R and C can be improved by 1; see [5].)…”

Section: Theorem 2 For Given M P There Exists An Isometric Embeddingmentioning

confidence: 99%

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Isometric embeddings of finite-dimensional $\ell_p$-spaces over the quaternions

Lyubich

Shatalova

St. Petersburg Math. J.

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Abstract. The nonexistence of isometric embeddings m q → n p with p = q is proved. The only exception is q = 2, p ∈ 2N, in which case an isometric embedding exists if n is sufficiently large, n ≥ N (m, p). Some lower bounds for N (m, p) are obtained by using the equivalence between the isometric embeddings in question and the cubature formulas for polynomial functions on projective spaces. Even though only the quaternion case is new, the exposition treats the real, complex, and quaternion cases simultaneously.

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Section: §1 Introductionmentioning

confidence: 84%

“…Another proof given in [25] can be extended to the complex case; see [23]. This is impossible to do in the quaternion case for the reason mentioned above.…”

Section: In Other Words Any Function φ ∈ φ K (M P) Is a Linear Combmentioning

confidence: 99%

Section: Theorem 1 If There Exists An Isometric Embeddingmentioning

confidence: 99%

“…Here n = p/2 + 1 because the centrally symmetric nodes can be joined. Moreover, this value of n turns out to be minimal [25,29].…”

Section: Theorem 1 If There Exists An Isometric Embeddingmentioning

confidence: 99%

“…needs a small modification of that argument; see [25,29]. (Note that (1.12) for K = R and C can be improved by 1; see [5].)…”

Section: Theorem 2 For Given M P There Exists An Isometric Embeddingmentioning

confidence: 99%

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Isometric embeddings of finite-dimensional $\ell_p$-spaces over the quaternions

Lyubich

Shatalova

St. Petersburg Math. J.

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Cubature Formulas, Geometrical Designs, Reproducing Kernels, and Markov Operators

Harpe

Pache

2005

Infinite Groups: Geometric, Combinatorial and Dynamical Aspects

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Abstract. Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way, several known results for spheres in Euclidean spaces, involving cubature formulas for polynomial functions and spherical designs, are shown to generalize to large classes of finite measure spaces (Ω, σ) and appropriate spaces of functions inside L 2 (Ω, σ). The last section points out how spherical designs are related to a class of reflection groups which are (in general dense) subgroups of orthogonal groups.Let (Ω, σ) be a finite measure space and let F be a vector space of integrable real-valued functions on Ω. It is a natural question to ask when and how integrals Ω ϕ(ω)dσ(ω) can be computed, or approximated, by sums x∈X W (x)ϕ(x), where X is a subset of Ω and W : X −→ R * + a weight function, for all ϕ ∈ F . When Ω is an interval of the real line, this is a basic problem of numerical integration with a glorious list of contributors: Newton (1671), Cotes (1711), Simpson (1743), Gauss (1814), Chebyshev (1874), Christoffel (1877), and Bernstein (1937), to quote but a few; see for some historical notes, , and . The theory is inseparable of that of orthogonal polynomials .When Ω is a space of larger dimension, the problems involved are often of geometrical and combinatorial interest. One important case is that of spheres in Euclidean spaces, with rotation-invariant probability measure, as in and . Work related to integrations domains with dim(Ω) > 1 goes back to and ; see also the result of Voronoi (1908) recalled in Item 1.15 below. Examples which have been considered include various domains in Euclidean spaces (hypercubes, simplices, ..., Item 1.20), surfaces (e.g. tori), and Euclidean spaces with Gaussian measures (Item 3.11).There are also interesting cases where Ω itself is a finite set .Section 1 collects the relevant definitions for the general case (Ω, σ). It reviews several known examples on intervals and spheres.Our main point is to show that there are two notions which are convenient for the study of cubature formulas, even if they are rarely explicitely introduced in papers of this subject.First, we introduce in Section 2 the formalism of reproducing kernel Hilbert spaces , , ), which is appropriate for generalizing to other spaces 1991 Mathematics Subject Classification. Primary 05B30, 65D32. Secondary 46E22.

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