EQUIVALENT CRITERION OF HAAR AND FRANKLIN SYSTEMS IN SYMMETRICAL SPACES

In the present article it is proved that if the Haar and Franklin syst ems are equivalent in a separable symmetric space E, the following con dition holds: 0<alpha(E)less-than-or-equal-to beta(E)<1, (1) where alp ha(E) and beta(E) are the Boyd indices of the space E, It is already k nown that if condition (1) is fulfilled, it follows that the Haar and Franklin systems are equivalent in the space E. Thereby, this estabish es that condition (1) is necessary and sufficient for the equivalence of the Haar and Franklin systems in E. In proving the assertion a numb er of interesting constructions involving Haar and Franklin polynomial s are presented and upper and lower bounds on the Franklin functions a pplied.