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Mat. Zametki, 1992, Volume 52, Issue 3, Pages 96–101 (Mi mz4704)  

Equivalent criterion of Haar and Franklin systems in symmetric spaces

I. Ya. Novikov

Research Institute for Mathematics State University of Voronezh

Abstract: In the present article it is proved that if the Haar and Franklin systems are equivalent in a separable symmetric space $E$, the following condition holds:
\begin{equation} 0<\alpha_E\leqslant\beta_E<1, \end{equation}
where $\alpha_E$ and $\beta_E$ are the Boyd indices of the space $E$. It is already known that if condition (1) is fulfilled, it follows that the Haar and Franklin systems are equivalent in the space $E$. Thereby, this estabishes that condition (1) is necessary and sufficient for the equivalence of the Haar and Franklin systems in $E$.
In proving the assertion a number of interesting constructions involving Haar and Franklin polynomials are presented and upper and lower bounds on the Franklin functions applied.

Full text: PDF file (700 kB)

English version:
Mathematical Notes, 1992, 52:3, 943–947

Bibliographic databases:

UDC: 517.5
Received: 09.03.1992

Citation: I. Ya. Novikov, “Equivalent criterion of Haar and Franklin systems in symmetric spaces”, Mat. Zametki, 52:3 (1992), 96–101; Math. Notes, 52:3 (1992), 943–947

Citation in format AMSBIB
\Bibitem{Nov92}
\by I.~Ya.~Novikov
\paper Equivalent criterion of Haar and Franklin systems in symmetric spaces
\jour Mat. Zametki
\yr 1992
\vol 52
\issue 3
\pages 96--101
\mathnet{http://mi.mathnet.ru/mz4704}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1194132}
\zmath{https://zbmath.org/?q=an:0795.42017}
\transl
\jour Math. Notes
\yr 1992
\vol 52
\issue 3
\pages 943--947
\crossref{https://doi.org/10.1007/BF01209614}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1992LF91500010}


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