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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
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.
📌 3 Key Takeaways
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Objective & Hypothesis:We consider Hill's equation y'' +($lambda$ -q)y=0 where $qin L^1[0,pi]$.
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Key Findings & Data:We consider Hill's equation y'' +($lambda$ -q)y=0 where $qin L^1[0,pi]$.
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Conclusion & Impact: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.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
We consider Hill's equation y'' +($lambda$ -q)y=0 where $qin L^1[0,pi]$. 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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Cite This Article
ÇOŞKUN, H. (2022). On the asymptotics of Fourier coefficients for the potential in Hill's equation. Turkish Journal of Mathematics, Cilt: 24 - Sayı: 1, 15–24. https://ajindex.com/articles/608912