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Register a new userAsymptotic gauges: generalization of Colombeau type algebras
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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.
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In [11], we introduced the notion of asymptotic gauge (AG), and we used it to construct Colombeau AG-algebras. This construction concurrently generalizes that of many different algebras used in Colombeaus theory, e.g. the special one $mathcal{G}^{srm}$, the full one $gse$, the NSA based algebra of asymptotic functions $hat{mathcal{G}}$, and the diffeomorphism invariant algebras $gsd$, $mathcal{G}^{2}$ and $hat{mathcal{G}}$. In this paper we study the categorical properties of the construction of Colombeau AG-algebras with respect to the choice of the AG, and we show their consequences regarding the solvability of generalized ODE.
We define a general notion of set of indices which, using concepts from pre-ordered sets theory, permits to unify the presentation of several Colombeau-type algebras of nonlinear generalized functions. In every set of indices it is possible to generalize Landaus notion of big-O such that its usual properties continue to hold. Using this generalized notion of big-O, these algebras can be formally defined the same way as the special Colombeau algebra. Finally, we examine the scope of this formalism and show its effectiveness by applying it to the proof of the pointwise characterization in Colombeau algebras.
This article is the natural continuation of the paper: Mukhammadiev A.~et al Supremum, infimum and hyperlimits of Colombeau generalized numbers in this journal. Since the ring $tilde{R}$ of Robinson-Colombeau is non-Archimedean, a classical series $sum_{n=0}^{+infty}a_{n}$ of generalized numbers $a_{n}intilde{R}$ is convergent if and only if $a_{n}to0$ in the sharp topology. Therefore, this property does not permit us to generalize several classical results, mainly in the study of analytic generalized functions (as well as, e.g., in the study of sigma-additivity in integration of generalized functions). Introducing the notion of hyperseries, we solve this problem recovering classical examples of analytic functions as well as several classical results.
Generalizations of Ostrowski type inequality for functions of Lipschitzian type are established. Applications in numerical integration and cumulative distribution functions are also given.
We classify the weak*-closed maximal left ideals of the measure algebra $M(G)$ for certain Hermitian locally compact groups $G$ in terms of the irreducible representations of $G$ and their asymptotic properties. In particular, we obtain a classification for connected nilpotent Lie groups, and the Euclidean rigid motion groups. We also prove a version of this result for certain weighted measure algebras. We apply our classification to obtain an analogue of Barnes Theorem on integrable representations for representations vanishing at infinity. We next study the relationship between weak*-closedness and finite generation, proving that in many cases $M(G)$ has no finitely-generated maximal left ideals. We also show that the measure algebra of the 2D Euclidean rigid motion group has a weak*-closed maximal left ideal that is not generated by a projection, and investigate whether or not it has any weak*-closed left ideals which are not finitely-generated.
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