Some inverse results for Hill's equation

AbstractWe consider Hill's equation y″+(λ−q)y=0 where q∈L1[0,π]. We show that if ln—the length of the n-th instability interval—is of order O(n−(k+2)) then the real Fourier coefficients ank,bnk of q(k)—k-th derivative of q—are of order O(n−2), which implies that q(k) is absolutely continuous almost everywhere for k=0,1,2,…