Spherical Functions on Euclidean Space

We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space $E^n = G/K$ where $G$ is the semidirect product $R^n \cdot K$ of the translation group with a closed subgroup $K$ of the orthogonal group O(n). We give exact parameterizations of the space of $(G,K)$--spherical functions by a certain affine algebraic variety, and of the positive definite ones by a real form of that variety. We give exact formulae for the spherical functions in the case where $K$ is transitive on the unit sphere in $E^n$.Comment: 10 page

Stepwise Square Integrability for Nilradicals of Parabolic Subgroups and Maximal Amenable Subgroups

In a series of recent papers we extended the notion of square integrability, for representations of nilpotent Lie groups, to that of stepwise square integrability. There we discussed a number of applications based on the fact that nilradicals of minimal parabolic subgroups of real reductive Lie groups are stepwise square integrable. Here, in Part I, we prove stepwise square integrability for nilradicals of arbitrary parabolic subgroups of real reductive Lie groups. This is technically more delicate than the case of minimal parabolics. We further discuss applications to Plancherel formulae and Fourier inversion formulae for maximal exponential solvable subgroups of parabolics and maximal amenable subgroups of real reductive Lie groups. Finally, in Part II, we extend a number of those results to (infinite dimensional) direct limit parabolics. These extensions involve an infinite dimensional version of the Peter-Weyl Theorem, construction of a direct limit Schwartz space, and realization of that Schwartz space as a dense subspace of the corresponding $L^2$ space.Comment: The proof of Theorem 5.9 is improved, several statements are clarified, and a certain number of typographical errors are correcte

Making automated computer program documentation a feature of total system design

It is pointed out that in large-scale computer software systems, program documents are too often fraught with errors, out of date, poorly written, and sometimes nonexistent in whole or in part. The means are described by which many of these typical system documentation problems were overcome in a large and dynamic software project. A systems approach was employed which encompassed such items as: (1) configuration management; (2) standards and conventions; (3) collection of program information into central data banks; (4) interaction among executive, compiler, central data banks, and configuration management; and (5) automatic documentation. A complete description of the overall system is given