The bilateral Laguerre transform

摘要

In the earlier paper [13], the crucial role of the complex plane in the formulation of the algorithms was evident, even though the algorithms were entirely in the real domain. For the bilateral transform, the complex plane is again very much present, with Laurent expansions, bilateral Laplace transformation, and conformal mapping entering as crucial tools. The first section extends the earlier formalism to the full continuum. That this extension is natural, and not just an artificial piecing together of the formalism for each half line, will be clear from (1.9), (1.12), and (1.13). The harmony of the basis will also emerge vividly in Sec. 3, which deals with the extent of the transform coefficients and associated uncertainty relations. The topic of extent is crucial to the utility of the Laguerre-transform method as a numerical tool. Numerical examples are presented in Sec. 5.