Mathematics

On the Haar and Franklin series with identical coefficients

G. G. Gevorkyan

Yerevan State University

Abstract: If $\big\lbrace \chi_n(x)\big\rbrace^\infty_{n=1}$ is Haar system and $\big\lbrace f_n(x)\big\rbrace^\infty_{n=0}$ is Franklin system, then for every $\lbrace a_n\rbrace^\infty_{n=0}$ and $p>0$ the following relation is proved
\begin{equation} \left\Vert\left\lbrace\sum\limits^\infty_{n=0}a^2_n f^2_n(x)\right\rbrace^{\frac{1}{2}} \right\Vert_p \sim \left\Vert\left\lbrace\sum\limits^\infty_{n=0}a^2_n \chi^2_{n+1}(x)\right\rbrace^{\frac{1}{2}} \right\Vert_p, \end{equation}

(1) has been proved in [2] when $p>l$ and in [4] when $\dfrac{1}{2}<p<l,$ but the methods of [2] and [4] are not applicable in the case $0<p\leq \dfrac{1}{2}$.
Some consequences are received from (1) as well.

UDC: 517.51

Received: 04.05.1989
Accepted: 05.10.1989