EQUIVALENT CRITERION OF HAAR AND FRANKLIN SYSTEMS IN SYMMETRICAL SPACES

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: 0<alpha(E)less-than-or-equal-to beta(E)<1, (1) 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.