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The Colombeau generalized nonlinear analysis and the Schwartz linear distribution theory. (English) Zbl 0951.46018
Author’s abstract: We adopt the elementary definition of the Colombeau algebra which was developed by J. Aragona and J. F. Colombeau [J. Math. Anal. Appl. 110, 179-199 (1985; Zbl 0596.46028)] and J. F. Colombeau [“Elementary introduction to new generalized functions”, North-Holland Math. Studies 113, Amsterdam (1985; Zbl 0584.46024)]. Unlike these works, we define an imbedding of the space $$C(\Omega)$$ of continuous functions, and then also the space of distributions $${\mathcal D}(\Omega)$$, into the algebra $${\mathcal G}(\Omega)$$ somewhat differently: our definition is close to that used by M. Oberguggenberger [“Multiplications of distributions and applications to partial differential equations”, Pitman Research Notes Math., Vol. 259, New York (1992; Zbl 0818.46036)] and is based on the elementary concept of convolution of a continuous function and a smooth function with compact support.…
The present paper is divided into nine sections. The material of Section 1 is classical: it is shown that the space of smooth functions with compact supports (i.e., test functions) is large enough, and some properties of the convolution of functions are recalled. In Section 2, we define the Colombeau algebra of generalized functions on an open set in $$\mathbb{R}^n$$ and establish its main properties. In Section 3, we introduce an algebra of generalized numbers so that Colombeau’s generalized functions assume values at individual points and can be integrated over compact sets. Also, in this section, we study solutions of algebraic equations within the framework of Colombeau’s theory. In Section 4, we define nonlinear operations of polynomial growth over generalized functions, composition of generalized functions, and restriction of generalized functions to linear subspaces. In Section 5, we present distributions which are defined as those generalized functions in Colombeau’s sense which locally (on every relatively compact open subset) can be represented as partial derivatives of continuous functions. Using the integration theory for generalized functions developed in Section 3, we obtain the classical formulation of distributions in a way that is accepted in Schwartz’ distribution theory. Then, in Section 6, we establish some of the classical properties of distributions which, in particular, allow us to display in Section 6.9 the natural character of the construction of the Colombeau algebra. The difficulties related to the problem of multiplication of distributions are described in Section 7, where, in particular, the Schwartz impossibility result is treated in more detail. As we have already mentioned, many classical operations (multiplication, composition, restriction, etc.) are necessarily changed in the algebra $${\mathcal G}(\Omega)$$, so, in Section 8, all these operations are recovered by means of the following two specific concepts: the equality in the sense of generalized distributions and the equality in the sense of the association. Finally, in Section 9, we present some further properties of the association: multiplication by the Dirac $$\delta$$ function, characterization of the product of distributions in the Colombeau algebra; also, the Heaviside generalized functions and the Dirac generalized functions are defined, and examples of discontinuous solutions to a first-order system in the conservative form are considered. In concluding our paper, we enumerate some recent papers (as known by the author) not mentioned in the body of this paper, which contribute to Colombeau’s theory and related theories of generalized functions.

MSC:
 46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.) 46F10 Operations with distributions and generalized functions
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References:
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