The orthogonal rational functions of Higgins and Christov and algebraically mapped Chebyshev polynomials

Abstract

It is shown that the rational functions of Higgins and Christov, orthogonal on [-[infinity], [infinity]], are Chebyshev polynomials of the first and second kinds with an algebraic change of variable. Because of these relationships, the existing theory and algorithms for mapped Chebyshev polynomials also apply to the rational functions: the Higgins and Christov functions have excellent numerical properties. However --precisely because of these same connections--it is usually simpler to use the change of variable rather than write computer programs that employ the Higgins and Christov functions themselves. Nonetheless, the result is a series of orthogonal rational functions. For some problems whose solutions decay slowly (algebraically rather than exponentially with |y|), such as the "Yoshida jet" in oceanography, a Christov expansion is the only spectral series that converges rapidly.