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 Trudy Mat. Inst. Steklova, 2003, Volume 243, Pages 237–243 (Mi tm431)

On the Gram Matrices of Systems of Uniformly Bounded Functions

B. S. Kashina, S. I. Sharekbc

a Steklov Mathematical Institute, Russian Academy of Sciences
b Université Pierre & Marie Curie, Paris VI
c Case Western Reserve University

Abstract: Let $A_N$, $N=1,2,…$, be the set of the Gram matrices of systems $\{e_j\}_{j=1}^N$ formed by vectors $e_j$ of a Hilbert space $H$ with norms $\|e_j\|_H\le 1$, $j=1,…,N$. Let $B_N(K)$ be the set of the Gram matrices of systems $\{f_j\}_{j=1}^N$ formed by functions $f_j\in L^\infty (0,1)$ with $\|f_j\|_{L^\infty (0,1)}\le K$, $j=1,…,N$. It is shown that, for any $K$, the set $B_N(K)$ is narrower than $A_N$ as $N\to \infty$. More precisely, it is proved that not every matrix $A$ in $A_N$ can be represented as $A=B+\Delta$, where $B\in B_N(K)$ and $\Delta$ is a diagonal matrix.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 243, 227–233

Bibliographic databases:
UDC: 517.5

Citation: B. S. Kashin, S. I. Sharek, “On the Gram Matrices of Systems of Uniformly Bounded Functions”, Function spaces, approximations, and differential equations, Collected papers. Dedicated to the 70th birthday of Oleg Vladimirovich Besov, corresponding member of RAS, Trudy Mat. Inst. Steklova, 243, Nauka, MAIK «Nauka/Inteperiodika», M., 2003, 237–243; Proc. Steklov Inst. Math., 243 (2003), 227–233

Citation in format AMSBIB
\Bibitem{KasSza03} \by B.~S.~Kashin, S.~I.~Sharek \paper On the Gram Matrices of Systems of Uniformly Bounded Functions \inbook Function spaces, approximations, and differential equations \bookinfo Collected papers. Dedicated to the 70th birthday of Oleg Vladimirovich Besov, corresponding member of RAS \serial Trudy Mat. Inst. Steklova \yr 2003 \vol 243 \pages 237--243 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm431} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2054436} \zmath{https://zbmath.org/?q=an:1084.46009} \transl \jour Proc. Steklov Inst. Math. \yr 2003 \vol 243 \pages 227--233 

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