72,714 research outputs found
Spherical Functions on Euclidean Space
We study special functions on euclidean spaces from the viewpoint of
riemannian symmetric spaces. Here the euclidean space where is
the semidirect product of the translation group with a closed
subgroup of the orthogonal group O(n). We give exact parameterizations of
the space of --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 is
transitive on the unit sphere in .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 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It is pointed out that in large-scale computer software systems, program documents are too often fraught with errors, out of date, poorly written, and sometimes nonexistent in whole or in part.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
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