0. Harmonic Analysis on the Euclidean Plane 5 On the other hand, by the spectral decomposition (classical Fourier series expansion) K(z, w) = ϕ λ(ϕ) ϕ(z) ϕ(w) the trace is given by Trace L = ϕ λ(ϕ) = u,v∈Z ˆ(u, v). Comparing both results, we get the trace formula m,n∈Z k(m, n) = u,v∈Z ˆ(u, v), which is better known as the Poisson summation formula. By a linear change of variables this formula can be modified for sums over general lattices Λ R2. On both sides of the trace formula on the torus Λ\R2 the terms are of the same type because the geometric and the spectral points range over dual lattices. However, one loses the self-duality on negatively curved surfaces, although the relevant trace formula is no less elegant (see Theorem 10.2). In particular, for a radially symmetric function the Poisson summation becomes Theorem (Hardy-Landau, Voronoi). If k C∞(R), 0 then =0 r( ) k( ) = =0 r( ) ˜( ) , where r( ) denotes the number of ways to write as the sum of two squares, r( ) = # (m, n) Z2 : m2 + n2 = , and ˜ is the Hankel type transform of k given by ˜( ) = π +∞ 0 k(t) J0(2π t) dt . Note that the lowest eigenvalue λ(1) = 4π2 with = 0 for the constant eigenfunction ϕ = 1 contributes ˜(0) = π +∞ 0 k(t) dt,
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