We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of the Colombeau simplified model. This generalizes the notion of G ∞ -regularity introduced by M. Oberguggenberger. A key point is that these regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to microanalysis of singularities of generalized functions, with respect to these regularities. We present a complete study of this topic, including properties of the Fourier transform (exchange and regularity theorems) and relationship with classical theory, via suitable results of embeddings.

In analogy to the classical Schwartz kernel theorem, we show that a large class of linear mappings admits integral kernels in the framework of Colombeau generalized functions. To do this, we introduce new spaces of generalized functions with slow growth and the corresponding adapted linear mappings. Finally, we show that, in some sense, Schwartz' result is contained in our main theorem.  2005 Elsevier Inc. All rights reserved.

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The aim of the work presented here is to connect two fields of functional analysis, on one hand the theory of sequence spaces and on the other hand the nonlinear theory of algebras of generalized functions, with the emphasis on the description of the latter. Associative differential algebras of generalized functions, containing the (embedded) delta distribution, with the ordinary product of continuous functions do not exist, as was proved by Schwartz [73]. But with the ordinary multiplication of smooth functions, such algebras do exist. One of the first and today most widely studied and used constructions has been introduced by Colombeau [8]. Nowadays, the theory of these so-called Colombeau type algebras is well-established and it is affirmed through many applications especially in nonlinear problems with strong singularities. Here we refer to the books [5,8,9,59,60,63] and to the numerous papers given in the references, while we apologize for all undue omissions. We also want to point out the progress made in the direction of PDE and differential geometry with applications in general relativity done by the DIANA group [24-27, 33, 34, 37, 38, 43, 45-47].On the other hand, sequence spaces of various type are a basic notion in investigations of various branches of functional analysis [48,49,50,51,52,53]. In this paper we show that Colombeau type algebras can be reconsidered as a class of sequence space algebras. We hope that our investigations in the field of generalized function algebras can serve as a motivation for those who are more interested in the functional analysis of sequence spaces.At the time when we started our work, the results of [24,25,26,27] related to the topology, and in general to functional analysis in the framework of Colombeau type generalized function algebras, were not known. Even now (five years later) they are not known properly. We would like to point out that this work significantly extends the well known theory relating to sharp topology. We will not give details about this work but advice the reader to consult the cited papers.The present paper extends our previous publications [13][14][15], where we elaborated separately on the general construction, on the issue of embeddings of distributions, ultradistributions and generalized hyperfunctions, and on functoriality and the different notions of association which we cast into a unified scheme, with new examples and developments relating to Maddox' sequence spaces, and sheaf theory.Colombeau constructed his well-known algebras by algebraic methods. No topology appeared in his construction. As we already mentioned, the different topologies and convergence structures defined on G appeared afterwards. Our first task in this paper [5]

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