Norm Dependence of the Coefficient Map on the Window Size1

For sparse exponent sequences (k) 1 ?1 , satisfying a suitablèseparation condition' deened by an auxiliary sequence , one has a `coeecient map' C giving (c k) 1 ?1 =: c from observation of f = P 1 k=?1 c k e ii k t on any arbitrarily small interval ?; ]. In terms of , we estimate the norm of C : L 2 ?; ] ! ` 2 , asymptotically as ! 0. In particular, for (k) 1 ?1 k p (p > 1) we get a bound which is exponential in (1==) 1=(p?1) , generalizing an earlier result for the case p = 2.