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Eduard A. Nigsch

Publications and source records attributed to Eduard A. Nigsch.

24 records · Page 2Linked to original sources
arXiv2016

The space $\dot{\mathcal{B}}'$ of distributions vanishing at infinity - duals of tensor products

Analogous to L.~Schwartz' study of the space $\mathcal{D}'(\mathcal{E})$ of semi-regular distributions we investigate the topological properties of the space $\mathcal{D}'(\dot{\mathcal{B}})$ of semi-regular vanishing distributions and give representations of its dual and of the scalar product with this dual. In order to determine the dual of the space of semi-regular vanishing distributions we generalize and modify a result of A. Grothendieck on the duals of $E \hat\otimes F$ if $E$ and $F$ are quasi-complete and $F$ is not necessarily semi-reflexive.

math.FA↗
arXiv2016

Some extensions to the functional analytic approach to Colombeau algebra

We extend the functional analytic approach to Colombeau-type spaces of nonlinear generalized functions in order to study algebras of tempered generalized functions. We obtain a definition of Fourier transform of nonlinear generalized functions which has a strict inversion theorem, agrees with the classical Fourier transform for tempered distributions and preserves well-known classical properties.

math.FA↗
arXiv2016

Nonlinear generalized sections of vector bundles

We present an extension of J. F. Colombeau's theory of nonlinear generalized functions to spaces of generalized sections of vector bundles. Our construction builds on classical functional analytic notions, which is the key to having a canonical geometric embedding of vector bundle valued distributions into spaces of generalized sections. This permits to have tensor products, invariance under diffeomorphisms, covariant derivatives and the sheaf property. While retaining as much compatibility to L. Schwartz' theory of distributions as possible, our theory provides the basis for a rigorous and general treatment of singular pseudo-Riemannian geometry in the setting of Colombeau nonlinear generalized functions.

math.FA↗
arXiv2015

On regularization of vector distributions on manifolds

One can represent Schwartz distributions with values in a vector bundle $E$ by smooth sections of $E$ with distributional coefficients. Moreover, any linear continuous operator which maps $E$-valued distributions to smooth sections of another vector bundle $F$ can be represented by sections of the external tensor product $E^* \boxtimes F$ with coefficients in the space $\mathcal{L}(\mathcal{D}', C^\infty)$ of operators from scalar distributions to scalar smooth functions. We establish these isomorphisms topologically, i.e., in the category of locally convex modules, using category theoretic formalism in conjunction with L. Schwartz' notion of $\varepsilon$-product.

math.FA↗