Bessel-like function approximation


Dean Douglass
03-12-2003, 05:51 PM
I would like advice in regard to computation of a Fourier integral, i^-n int(e^(itx) U sub n(x),x,0,1), for a wide range of n and t. Owing to the extreme behavior of the Chebychev polynomial of the 2nd kind, U sub n, near x=1, an uncorrected DFT is entirely inadequate. The integral satisfies a nonhomogeneous Bessel recursion relation. Upward recursion would be highly desirable, but I cannot any info. on the numerical stability, up or down, of the nonhomogeneous relation. Is this because it is well-known to be stable, or unstable? Because the auxillary recursion relation associated with the Clenshaw summation (Numerical Recipes in C, p. 182) has the same functional form for homo- & nonhomo. recursion relations, it appears readily adaptable, but relatively inefficient. I have no limiting case except t = 0 for testing, so it is difficult to have faith in numerical experimentation in the absence of insight. The benefit of someone's experience in a similar context is earnestlt sought.