Inner functions and bicyclic vectors

We consider weights omega on Z such that omega (n) --> 0 as n --> +infinity , omega (n) --> +infinity as n --> -infinity, and satisfying some regularit y conditions. Set [GRAPHICS] and denote by S-omega: (u(n))(n is an element ofZ) -->(u(n-1))(n is an elem ent ofZ) the usual shift on l(omega)(2). We show that if [GRAPHICS] then there exists a singular inner function U such that (U) over cap = ((U) over cap (n))(n greater than or equal to0) is not bicyclic in l(omega)(2), that is, the closure of Span(S-omega(n)(U) over cap: n is an element of Z} is a proper subspace of l(omega)(2).