Special Session 164: Periodic and Ergodic Schrodinger Operators
The Fourier Ratio and Chang`s Lemma
Giovanni Garza
University of Delaware USA
Co-Author(s): K. ALDALEH, W. BURSTEIN, G. GARZA, G. HART, A. IOSEVICH, J. IOSEVICH, A. KHALIL, J. KING, N. KULKARNI, T. LE, I. LI, A. MAYELI, B. MCDONALD, K. NGUYEN, AND N. SHAFFER
Abstract:
Given a function $f:\Z_N\to\C$, we denote by $\widehat{f}$ its Fourier transform which is given by
\begin{equation*}
\widehat{f}(m)=\frac{1}{\sqrt{N}}\sum_{x\in\Z_N}e^{-2\pi ixm/N}f(x).
\end{equation*}
We introduce the function $FR(f)=\frac{\norm{\widehat{f}}_1}{\norm{\widehat{f}}_2}$, and examine how this ratio tells us the extent to which we can approximate $f$ by a low degree trigonometric polynomial. Finally, we prove a generalized version of Chang`s lemma and show that the sparse spectrum approximation of $f$ has some additive structure.