SYMMETRY REDUCTION OF FOURIER KERNELS

Fourier transforms of functions of several variables invariant under c ertain symmetry groups are studied, with particular reference to funct ions f(x(1)... x(N)) of N three-component vectors invariant under rigi d rotations. Here we use symmetry to enhance the efficiency of evaluat ion of the integrals. The Fourier transform can be written as an integ ral F (k) = integral d mu(x)K(k,x)f(x) over rotationally invariant qua ntities x. The kernel K, the average of exp(i Sigma k(i).x(i)) over th e rotation group SO(3), is reduced to a single integral, integral(0)(1 ) J(0)(1/2(A(xx)+A(yy))u)J(0)(1/2(A(xx)-A(yy))(1-u)) exp(iA(zz)(2u - 1 )) du, a function of the eigenvalues of the dyadic A = Sigma(i=1)(N)k( i)x(i). For N = 1 the familiar Hankel transform is recovered. For N = 2 the kernel reduces to a single integral of elementary functions, equ al to the local spin-flip propagator in a one-dimensional tight-bindin g antiferromagnet. A variety of forms is given, and useful asymptotic forms are found in Various limits. Recent numerical methods for the ev aluation of irregular oscillatory integrals are applied to the kernel in the N = 2 case. (C) 1998 Academic Press.