Über homogene Polynome in ($L^{2}$)

In this paper we discuss the notion of singular vector tuples of a complex valued d-mode tensor of dimension m 1 × . . . × m d . We show that a generic tensor has a finite number of singular vector tuples, viewed as points in the corresponding Segre product. We give the formula for the number of singular vector tuples. We show similar results for tensors with partial symmetry. We give analogous results for the homogeneous pencil eigenvalue problem for cubic tensors, i.e. m 1 = . . . = m d . We show uniqueness of best approximations for almost all real tensors in the following cases: rank one approximation; rank one approximation for partially symmetric tensors (this approximation is also partially symmetric); rank-(r 1 , . . . , r d ) approximation for d-mode tensors.2010 Mathematics Subject Classification. 14D21, 15A18, 15A69, 65D15, 65H10, 65K05.

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“…Actually, this result is equivalent to Banach's theorem [1]. See [23] for another proof of Banach's theorem.…”

Section: Introductionmentioning

confidence: 90%

The Number of Singular Vector Tuples and Uniqueness of Best Rank-One Approximation of Tensors

2014

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“…Note that inequalities (9) are sharp in the general case. However, Banach [16] showed that the norms of A α1,...,α l and σÃ α1,...,α l coincide for p = 2. Inequality (10) implies in particular that the operator σÃ α1,...,α l acts in L p if so doesÃ α1,...,α l ; the converse is not necessarily valid.…”

Section: Relationsmentioning

confidence: 97%

New existence theorems for Lyapunov-Schmidt integral equations

Zabreiko

Shirokanova

2004

Diff Equat

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“…A famous result, studied by Banach [5] and many other authors, for example [10], [11], [18], [20], [21], [29], asserts that for every continuous symmetric n-linear form L : H × · · · × H → K, K = R or C, we have L = L , where L is the continuous homogeneous polynomial of degree n associated to L. If X is a Banach space over K, we recall that a continuous n-homogeneous polynomial L : X → K is, by definition, the restriction to the diagonal of a necessarily unique symmetric continuous n-linear form L : X × · · · × X → K; that is L (x) = L (x, . .…”

Section: Introduction and Notationmentioning

confidence: 99%

Homogeneous polynomials and extensions of Hardy‐Hilbert's inequality

Anagnostopoulos

Sarantopoulos

Tonge

2011

Mathematische Nachrichten

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If L is a continuous symmetric n-linear form on a real or complex Hilbert space and L is the associated continuous n-homogeneous polynomial, then L = L . We give a simple proof of this well-known result, which works for both real and complex Hilbert spaces, by using a classical inequality due to S. Bernstein for trigonometric polynomials. As an application, an open problem for the optimal lower bound of the norm of a homogeneous polynomial, which is a product of linear forms, is related to the so-called permanent function of an n × n positive definite Hermitian matrix. We have also derived generalizations of Hardy-Hilbert's inequality.

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Über homogene Polynome in ($L^{2}$)

Cited by 85 publications

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The Number of Singular Vector Tuples and Uniqueness of Best Rank-One Approximation of Tensors

The Number of Singular Vector Tuples and Uniqueness of Best Rank-One Approximation of Tensors

New existence theorems for Lyapunov-Schmidt integral equations

Homogeneous polynomials and extensions of Hardy‐Hilbert's inequality

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