Abstract
We present a construction of algebras of generalized functions of Colombeau-type which, instead of asymptotic estimates with respect to a regularization parameter, employs only topological estimates on certain spaces of kernels for its definition.
Highlights
Colombeau algebras, as introduced by Colombeau [1,2], today represent the most widely studied approach to embedding the space of Schwartz distributions into an algebra of generalized functions such that the product of smooth functions as well as partial derivatives of distributions are preserved
For an overview we refer to works on asymptotic scales [3,7], (C, E, P)-algebras [5], sequence spaces with exponent weights [6] and asymptotic gauges [8]
In this article we will present an algebra of generalized functions which instead of asymptotic estimates employs only topological estimates on certain spaces of kernels for its definition
Summary
As introduced by Colombeau [1,2], today represent the most widely studied approach to embedding the space of Schwartz distributions into an algebra of generalized functions such that the product of smooth functions as well as partial derivatives of distributions are preserved. In this article we will present an algebra of generalized functions which instead of asymptotic estimates employs only topological estimates on certain spaces of kernels for its definition. This is a direct generalization of the usual seminorm estimates valid for distributions. We give canonical mappings into the most important Colombeau algebras, which points to a certain universality of the construction offered here
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