A SIMPLE WILSON ORTHONORMAL BASIS WITH EXPONENTIAL DECAY

Following a basic idea of Wilson ["Generalized Wannier functions," preprint] orthonormal bases for L2(R) which are a variation on the Gabor scheme are constructed. More precisely, phi is-a-member-of L2(R) is constructed such that the psi-ln, l is-a-member-of N, n is-a-member-of Z, defined by psi-on(x) = phi(x-n) psi-ln(x) = square-root 2-phi (x-n/2) cos (2-pi-lx) if 1 not-equal 0, l + n is-a-member-of 2Z = square-root 2-phi (x-n/2) sin (2-pi-lx) if l not-equal 0, l +n is-a-member-of 2Z +1, constitute an orthonormal basis. Explicit examples are given in which both phi and its Fourier transform phi have exponential decay. In the examples phi is constructed as an infinite superposition of modulated Gaussians, with coefficients that decrease exponentially fast. It is believed that such orthonormal bases could be useful in many contexts where lattices of modulated Gaussian functions are now used.