4 0. Harmonic Analysis on the Euclidean Plane is just the spectral resolution of D on functions satisfying proper decay conditions. Another view of this matter is offered by invariant integral operators (Lf)(z) = R2 k(z, w) f(w) dw. For L to be G-invariant it is necessary and sufficient that the kernel function, k(z, w), depends only on the difference z w, i.e. k(z, w) = k(z w). Such an L acts by convolution: Lf = k f. One shows that the invariant integral operators mutually commute and that they commute with the Laplace op- erator as well. Therefore the spectral resolution of D can be derived from that for a sufficiently large family of invariant integral operators. By direct computation one shows that the exponential function ϕ(x, y) = e(ux+vy) is an eigenfunction of L with eigenvalue λ(ϕ) = ˆ(u, v), the Fourier transform of k(z). Of particular interest will be the radially symmetric kernels: k(x, y) = k(x2 + y2) , k(r) C∞(R+). 0 Using polar coordinates, one finds that the Fourier transform is also radially symmetric. More precisely, ˆ(u, v) = π +∞ 0 k(r) J0( λr) dr, where λ = 4π2(u2 + v2) and J0(z) is the Bessel function J0(z) = 1 π π 0 cos(z cos α) dα. Classical analytic number theory benefits a lot from harmonic analysis on the torus Z2\R2 (which is derived from that on the free space R2 by the unfolding technique), as it exploits properties of periodic functions. Restricting the domain of the invariant integral operator L to periodic functions, we can write (Lf)(z) = Z2\R2 K(z, w) f(w) dw, where K(z, w) = p∈Z2 k(z + p, w) , by folding the integral. Hence the trace of L on the torus is equal to Trace L = Z2\R2 K(w, w) dw = p∈Z2 k(p) = m,n∈Z k(m, n).
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