Lecture 4: Covering the Large Spectrum via Dual-sparse Approximation

1 Discrete Fourier analysis In this lecture, we use the dual-sparse approximation theorem from the last lecture to prove some results in discrete Fourier analysis. For simplicity, we restrict ourselves to the setting of G ¿ n 2 , but the theorems hold (when suitably restated) for any finite abelian group G. Fourier analysis over ¿ n 2. We use ¿ 2 {0, 1} to denote the field on two elements. Let G ¿ n 2 be equipped with the uniform measure µ. We usê G ¿ n 2 to denote the dual group (though we use the notations G andˆG to distinguish primal and dual objects). We will use the definitions from Lecture 3 (Section 3).