On the Haar and Franklin series with identical coefficients

G. G. Gevorkyan

Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (1989), pp. 3-9

Résumé

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 $\dfrac{1}{2}$ but the methods of [2] and [4] are not applicable in the case $0$. Some consequences are received from (1) as well.