@MISC{Bell10uncertaintyprinciples, author = {Jordan Bell}, title = {Uncertainty principles and . . .}, year = {2010} }

Share

OpenURL

Abstract

In this paper I am careful to distinguish between the space of signals and the space of their Fourier transforms. Folland and Sitaram [4] give a survey of uncertainty principles in analysis. First we will go through the talk by Emmanuel Candès and Terence Tao, “The uniform uncertainty principle and compressed sensing”, Harmonic Analysis and Related Topics, Seville, December 5, 2008. Let G = Z/nZ. Let L(G) denote the set of functions from G to C. For T ⊆ G, let L(T) be the set of functions from G to C whose support is contained in T. For x ∈ G, define the Dirac delta function δx: G → C centered at x by 1, y = x δx(y) = 0, y = x. The set {δx}x∈G is a basis for the vector space L(G). For f ∈ L(G), f(y) = ∑ f(x)δx(y), y ∈ G. x∈G We define an inner product on L(G) by 〈f, g〉G = 1 ∑ f(x)g(x), |G| x∈G f, g ∈ L(G). The L2 norm on L(G) is given by ‖f‖L2 (G) = 〈f, f 〉 1/2