## Project Details

### Abstract

Wider research context / theoretical framework: It is a well-known problem in the generalized functions community that the Fourier transform of Colombeau generalized functions, defined using damping measures, lacks important results such as the Fourier inversion theorem. Another long-standing open problem is the seeming impossibility to prove the Cauchy-Kowalevski theorem without growth restrictions of log type.

Hypotheses/research questions/objectives: The main goals we want to develop in the present proposal are as follows:

1. Hyperfinite Fourier transform, which is applicable to all generalized smooth functions and for which we already proved the inversion formula in the 1-dimensional case at all finite points;

2. Classical Fourier transform, having better formal properties with respect to the former and applicable to all Colombeau generalized functions, not only those of tempered type;

3. To prove the Cauchy-Kowalevski theorem for hyper-analytic generalized smooth functions without growth restrictions and applicable at least to all Sobolev-Schwartz distributions.

Approach/methods: The nonlinear theory of generalized smooth functions will be the main setting. This theory recently emerged as a minimal extension of Colombeau's theory that allows for more general domains for generalized functions, resulting in the closure with respect to composition, a better behavior on unbounded sets and new general existence results for singular partial differential equations. This allows us to use the strong integration theory of generalized smooth functions on unbounded domains and the set of generalized natural numbers (called hypernatural numbers) for the notion of hyperseries.

Level of originality/innovation: We can show that the new notion of hyperseries is able to overcome several counter-intuitive properties also of other non-Archimedean settings, where it is well-known that a series converges if and only if its general term tends to zero. For example, we already proved the convergence of the exponential hyperseries for all finite points and extended all the classical convergence tests to hyperseries. The wide range of applicability of classical and hyperfinite Fourier transforms, with the full inversion theorem at all finite points, finds potential applications in quantum mechanics, signal analysis and solutions of differential equations.

Primary researchers involved: 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.

Hypotheses/research questions/objectives: The main goals we want to develop in the present proposal are as follows:

1. Hyperfinite Fourier transform, which is applicable to all generalized smooth functions and for which we already proved the inversion formula in the 1-dimensional case at all finite points;

2. Classical Fourier transform, having better formal properties with respect to the former and applicable to all Colombeau generalized functions, not only those of tempered type;

3. To prove the Cauchy-Kowalevski theorem for hyper-analytic generalized smooth functions without growth restrictions and applicable at least to all Sobolev-Schwartz distributions.

Approach/methods: The nonlinear theory of generalized smooth functions will be the main setting. This theory recently emerged as a minimal extension of Colombeau's theory that allows for more general domains for generalized functions, resulting in the closure with respect to composition, a better behavior on unbounded sets and new general existence results for singular partial differential equations. This allows us to use the strong integration theory of generalized smooth functions on unbounded domains and the set of generalized natural numbers (called hypernatural numbers) for the notion of hyperseries.

Level of originality/innovation: We can show that the new notion of hyperseries is able to overcome several counter-intuitive properties also of other non-Archimedean settings, where it is well-known that a series converges if and only if its general term tends to zero. For example, we already proved the convergence of the exponential hyperseries for all finite points and extended all the classical convergence tests to hyperseries. The wide range of applicability of classical and hyperfinite Fourier transforms, with the full inversion theorem at all finite points, finds potential applications in quantum mechanics, signal analysis and solutions of differential equations.

Primary researchers involved: 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.

Status | Not started |
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### Keywords

- Colombeau generalized functions
- non-Archimedean functional analysis
- Colombeau algebras
- nonlinear analysis
- Generalized functions