We plan to explore systematically a number of closely related problems, using recent techniques of Fourier analysis. Key topics are positive definite functions, uncertainty principles, and duality theory in, say, locally compact groups. Wide interest in an d current flourishing of Fourier analysis and in particular extremal problems stem from applications, ranging even to far fields of science, from mathematical areas as number theory, geometry and analysis, to applied mathematics including numerical analysi s, finite fast Fourier transforms and information theory, etc. Addressing concrete problems, we expect the results being useful also in signal processing, phase reconstruction, electric circuits, crystallography, coding, antenna design, radar engineering. We plan basic, fundamental research, but the work has also an interdisciplinary character, centered on applications in mathematics, physics and engineering. We will publish results in acknowledged international research journals. The project helps the res earcher to draw from his wide, strong background in mathematical analysis when focusing his work. It provides him intensive training from a distinguished French school of Fourier analysis, but also integrate him into activities of the respective European n etwork HARP, including continuation of existing collaboration with Fourier analysts from other branches. Moreover, the project aims at providing the scientific elevation of the researcher's work to allow him achieving the highest scientific degree at his b ase, in Hungary. That should help him to consolidate a new, strong research team in Hungary, where forming of a Fourier analysis group has already started. Members of the group, in particular young scientists, are planned to be involved considerably into t he activities of the project, using even other funding resources for visits. Long range cooperation is planned, even extending prospective continuation of the HARP network by a new Hungarian branch.
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