Symmetry reduction of Fourier kernels
The authors examine Fourier transforms of functions of several variables invariant under the group \(G= \text{SO}(d)\) of proper rigid rotations of the frame of \(N\) vectors \(\{x_i\}^N_{i= 1}\) in \(\mathbb{R}^d\). The Fourier transforms of these functions can be written as \[ F(k;t)= \int_{\mathbb{R}^{Nd}/G} K(k,x;t) f(x) d\mu(x), \] where \[ K(k,x;t)= \int_{g\in G} e^{it\sum^N_{i= 1} k_i\cdot gx_i} d\mu(g). \] In general, the kernel cannot be expressed in closed form in terms of elementary functions. The authors restrict their attention to the three-dimensional case \(d= 3\): They show that the above kernel reduces to \[ \int^1_0 J_0(t(A_{xx}+ A_{yy})u) J_0(t(A_{xx}+ A_{yy})(1- u)) e^{it A_{zz}(2u- 1)} du, \] where \(A_{zz}\leq A_{yy}\leq A_{xx}\) are the eigenvalues of the matrix \[ A= \Biggl(\sum^N_{i= 1} (k_i)_\mu(x_i)_\nu\Biggr)^3_{\nu, \mu=1}. \] For \(N= 2\), the kernel can be written as single integral of elementary functions. For \(N= 1\), the familiar Hankel transform is recovered. Finally, the authors apply recent numerical methods for the evaluation of irregular oscillatory integrals to the kernel in case \(N= 2\).
scientific article; zbMATH DE number 1290133