Summability methods for hermite functions

Abstract

Many problems in equatorial oceanography can be analytically solved via series of Hermite functions, but unfortunately these expansions converge very poorly. In this note, we describe two simple tricks for accurately evaluating such series. The first, due to Moore, is to apply numerical weighting factors to the last four terms in the series. This is a special case of a more powerful technique known as the `Euler-Abel' method which is almost as easy to apply. Our numerical examples show that both methods are very effective. The Euler-Abel method gives an error which decreases exponentially fast as N, the number of terms retained in the truncated series, increases. Moore's method only reduced the error by a factor of 0(1/N2) in comparison to the original series, but this is more than enough for most practical purposes, and this trick is simpler and distributes the error more uniformly in latitude than the Euler-Abel transformation. A conservative rule-of-thumb is that both methods give errors too small to observe on a graph on the range where N is the number of terms in the Hermite series.