Fourier transforms and Cauchy-Kowalevski theorem for GSF

The main aim of the proposed project is to develop some important results in the mathematical analysis of generalized smooth functions (GSF). These functions are useful to describe rapidly changing phenomena such as breaking of steel structures, switching of electrical circuits, motion through different or granular media, quantum mechanics, etc. In this setting, new general existence results for differential equations was recently proved. The main goals we want to develop in the present proposal are as follows: 1. Hyperfinite Fourier transform applicable to all GSF; 2. Classical Fourier transform applicable to rapidly decreasing GSF; 3. To prove the Cauchy-Kowalevski theorem for hyper-analytic generalized smooth functions. The wide range of applicability of these results finds potential applications in quantum mechanics, signal analysis and solutions of differential equations. Primary researchers involved are the applicant P. Giordano and Prof. M. Kunzinger will supervise the PhD candidates A. Mukhammadiev and D. Tiwari. The present project financially aims to support only the last year of their PhD studies.

The main aim of the project was to develop some important results in the mathematical analysis of generalized smooth functions (GSF). These functions are useful to describe rapidly changing phenomena such as breaking of steel structures, switching of electrical circuits, motion through different or granular media, quantum mechanics, etc. In this setting, new general existence results for differential equations was recently proved. The main goals we developed in the present proposal were as follows: 1. Hyperfinite Fourier transform applicable to all GSF; 2. Classical Fourier transform applicable to rapidly decreasing GSF; 3. A theory of generalized holomorphic functions, i.e. generalized functions of a complex variable. The wide range of applicability of these results finds potential applications in quantum mechanics, signal analysis and solutions of differential equations. Primary researchers involved have benn the applicant P. Giordano and Prof. M. Kunzinger and four PhD candidates. Two of them already successfully finished their PhD studies.