On the Asymptotics of Fourier Coefficients for the Potential in Hill's Equation

Authors: HASKIZ COŞKUN

Abstract: We consider Hill's equation $y'' +(\lambda -q)y=0$ where $q\in L^{1}[0,\pi ].$ We show that if $l_{n}-$the length of the $n-th$ instability interval$-$ is of order $O(n^{-k})$ then the real Fourier coefficients $a_{n},b_{n}$ of $q$ are of the same order for$(k=1,2,3)$, which in turn implies that $q^{(k-2)}$, the $(k-2)th$ derivative of $q$, is absolutely continuous almost everywhere for $k=2,3.$

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