Chapter 0 Harmonic Analysis on the Euclidean Plane We begin by presenting the familiar case of the euclidean plane R2 = (x, y) : x, y ∈ R . The group G = R2 acts on itself as translations, and it makes R2 a homo- geneous space. The euclidean plane carries the metric ds2 = dx2 + dy2 of curvature K = 0, and the Laplace-Beltrami operator associated with this metric is given by D = ∂2 ∂x2 + ∂2 ∂y2 . Clearly the exponential functions ϕ(x, y) = e(ux + vy) , (u, v) ∈ R2 , are eigenfunctions of D (D + λ)ϕ = 0 , λ = λ(ϕ) = 4π2(u2 + v2) . The well-known Fourier inversion ˆ(u, v) = f(x, y) e(ux + vy) dx dy , f(x, y) = ˆ(u, v) e(−ux − vy) du dv , 3 http://dx.doi.org/10.1090/gsm/053/02

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