On a Fourier integral over SO(3)

Abstract

A two-matrix function of general interest in the areas of configuration statistics of macromolecules, number theory, harmonic analysis, and multivariate statistics is studied. The function is defined as a Fourier integral over SO(3), the Lie group of orthogonal 3×3 matrices with unit determinant. This six-variable function is first expressed as a product of a three-variable function and an exponential function of an additional variable, thereby reducing the total number of independent variables by 2. The new function with three parameters is expressible either as a double integral or as a series in one of the variables with the coefficients being polynomials in the other two. A special, nontrivial case where one of three arguments of the function takes a particular value is explored thoroughly. The resulting two-variable function is real valued and is an oscillating function of one of the variables when the other is fixed. When this function is expanded as a power series in one of the two variables, it generates polynomials in the other variable. Numerical analysis of this series shows it to be rapidly convergent and it is of practical use in the numerical evaluation of the function. Although the connection between these newly found polynomials and zonal polynomials has not been investigated, the parametrization for the four new variables of the two-matrix function studied may well prove useful in the effective numerical evaluation of the function when expressed alternatively as a series in zonal polynomials with an exponential part factored out.