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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

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


Authors: Yu. I. Lyubich and O. A. Shatalova
Original publication: Algebra i Analiz, tom 16 (2004), nomer 1.
Journal: St. Petersburg Math. J. 16 (2005), 9-24
MSC (2000): Primary 46B04
Published electronically: December 14, 2004
MathSciNet review: 2068351
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Abstract: The nonexistence of isometric embeddings $\ell_{q}^{m}\to\ell_p^{n}$with $p\neq q$ is proved. The only exception is $q=2$, $p\in 2\mathbb{N} $, in which case an isometric embedding exists if $n$ is sufficiently large, $n\ge N(m,p)$. 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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Additional Information

Yu. I. Lyubich
Affiliation: Department of Mathematics, Technion, Haifa 32000, Israel
Email: lyubich@tx.technion.ac.il

O. A. Shatalova
Affiliation: Department of Mathematics, Technion, Haifa 32000, Israel
Email: oksana@tx.technion.ac.il

DOI: http://dx.doi.org/10.1090/S1061-0022-04-00842-8
PII: S 1061-0022(04)00842-8
Keywords: Isometric embeddings, cubature formulas, addition theorem
Received by editor(s): October 31, 2003
Published electronically: December 14, 2004
Dedicated: Dedicated to M. Sh. Birman on the occasion of his 75th birthday
Article copyright: © Copyright 2004 American Mathematical Society