Asymptotic coefficients of hermite function series

Abstract

By using complex variable methods (steepest descent and residues) to asymptotically evaluate the coefficient integrals, the numerical analysis of Hermite function series is discussed. There are striking similarities and differences with the author's earlier work on Chebyshev polynomial methods ( J. Comp. Phys. 45 (1982), 45–49) for infinite or semi-infinite domains. Like Chebyshev series, the Hermite coefficients are asymptotically given by the sum of two types of terms: (i) stationary point (steepest descent) contributions and (ii) residues at the poles of f( z), the function being expanded as a Hermite series. The stationary point term is determined solely by the asymptotic behavior of f( z), i.e., how rapidly f( z) decays as z → ∞ along the real axis. Unlike Chebyshev series, however, it is necessary to perform a separate analysis for functions which decay faster or slower than the Gaussian function exp[- Az 2 ]. Singular functions, too, fall into two categories. Those that decay rapidly with z have asymptotic Hermite coefficients which are dominated by the singularity, but functions which decay as slowly as sech( z) or slower have Hermite coefficients dominated by the stationary point terms, and the singularity is irrelevant. The end product of the analysis is the same as in the earlier work: simple, explicit formulas to optimize the efficiency of Hermite methods and estimate a priori how many degrees of freedom are needed, provided one knowns at least crudely: (i) the asymptotic behavior of f( z) and (ii) its singularity nearest the real axis. Rather surprisingly, one finds Hermite functions superior to Chebyshev polynomials for some classes of functions when the computational domain is infinite.