| Let theta be an inner function for the upper half plane. We are interested in the description of positive functions wx defined on R for which fx ⩽wx, x∈R, for some non-zero f in the model space Ktheta of H2( R ). In the special case theta(x) = e isigmax, sigma > 0, the celebrated multiplier theorem of Beurling and Malliavin gives an almost complete solution. We give a general multiplier theorem applicable to all Ktheta with theta having a smooth argument on the real line. This result applies, in particular, to the spaces KB formed from Blaschke products B for the upper half plane, meromorphic in C . We show that in this case the criterion for a function w to have the above property depends on the distribution of the zeros of B. |
