Multiplication and mixed differentiation in generalized Fock spaces
The classical distribution theory developed by L. Schwartz has an intrinsic multiplication problem. A new generalized function theory developed by \textit{J. Colombeau} [1. New generalized functions and multiplication of distributions (1984; Zbl 0532.46019), 2. Elementary introduction to new generalized functions (1985; Zbl 0584.46024)] provides a solution to the multiplication problem. \textit{T. Todorov} [Sequential approach to Colombeau's theory of generalized functions, ICTP IC126, Trieste, Italy (1987)] re-examined this new theory in the spirit of non-denumerable sequences, termed ultrapowers. In this paper, a few basic results in the new tempered distribution theory has been reviewed before considering the multiplication property in this setting by blending the Colombeau's and Todorov's sequential approach. The criteria to ensure the existence of functional derivatives have been developed. A principal application for the results developed will be modelling a physical system consisting of an `infinite' number of particles (bosons or fermions). Section 1 is the introduction. Section 2 deals with classical test spaces. Section 3 deals with classical generalized functions. Functional derivatives are considered in section 4. Sections 5 and 6 consider the scale of Fréchet spaces, \(\Gamma^{P^ B}\) and the space, \({\mathcal H}(\mathbb{R}^ k\times \mathbb{S}';\mathbb{R})\) respectively. Section 7 deals with the spaces of new tempered Fock functionals. Infinite order differential operators are considered in section 8. The author has claimed that several interesting transforms such as generalized Laplace or Stieltjes transform could be studied in the spaces developed by the author. This has, in fact, been initiated in this setting and will be the text of a forthcoming paper of the author with Takači.
scientific article; zbMATH DE number 62071
scientific article