In the 1950s L. Schwartz proved his famous impossibility result: for every k ∈ N there does not exist a differential algebra (A, +, ⊗, D) in which the distributions can be embedded, where D is a linear operator that extends the distributional derivative and satisfies the Leibnitz rule (namely D(u ⊗ v) = Du ⊗ v + u ⊗ Dv) and ⊗ is an extension of the pointwise product on C 0 (R).In this paper we prove that, by changing the requests, it is possible to avoid the impossibility result of Schwartz. Namely we prove that it is possible to construct an algebra of functions (A,+, ⊗, D) such that (1) the distributions can be embedded in A in such a way that the restriction of the product to C 1 (R) functions agrees with the pointwise product, namely for every f, g ∈ C 1 (R)and (2) there exists a linear operator D : A → A that extends the distributional derivative and satisfies a weak form of the Leibnitz rule.The algebra that we construct is an algebra of restricted ultrafunctions, which are generalized functions defined on a subset Σ of a nonarchimedean field K (with R ⊂ Σ ⊂ K) and with values in K. To study the restricted ultrafunctions we will use some techniques of nonstandard analysis. Mathematics subject classification: 13N99, 26E30, 26E35, 46F30.

Ultrafunctions are a particular class of functions defined on a non-Archimedean field R * ⊃ R. They provide generalized solutions to functional equations which do not have any solutions among the real functions or the distributions. In this paper we analyze sistematically some basic properties of the spaces of ultrafunctions.Mathematics subject classification: 26E30, 26E35, 46F30.

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Abstract. We use the general notion of set of indices to construct algebras of nonlinear generalized functions of Colombeau type. They are formally defined in the same way as the special Colombeau algebra, but based on more general "growth condition" formalized by the notion of asymptotic gauge. This generalization includes the special, full and nonstandard analysis based Colombeau type algebras in a unique framework. We compare Colombeau algebras generated by asymptotic gauges with other analogous construction, and we study systematically their properties, with particular attention to the existence and definition of embeddings of distributions. We finally prove that, in our framework, for every linear homogeneous ODE with generalized coefficients there exists a minimal Colombeau algebra generated by asymptotic gauges in which the ODE can be uniquely solved. This marks a main difference with the Colombeau special algebra, where only linear homogeneous ODEs satisfying some restriction on the coefficients can be solved.

This paper deals with a new kind of generalized functions, called "ultrafunctions", which have been introduced recently in [5] and developed in [10] and [11]. Their peculiarity is that they are based on a Non Archimedeanfield, namely on a field which contains infinite and infinitesimal numbers. Ultrafunctions have been introduced to provide generalized solutions to equations which do not have any solutions, not even among the distributions. Some applications of this kind will be presented in the second part of this paper