Fourier analysis and applications to completely integrable systems

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Do and his collaborators will use new methods to investigate several questions related to Fourier analysis and its applications to completely integrable systems. This project concerns convergence of Fourier series along `thin' sub-sequences of partial sums, as motivated by a conjecture of Konyagin; extensions of classical and non-classical estimates related to Fourier series to weighted settings, with motivations coming from work of Germain-Masmoudi-Shatah in partial differential equations; a new approach to Plancherel-Rotach asymptotics of orthogonal polynomials with rough varying weights, motivated by applications in spectral study of unitary random matrices and by applications in other non-commutative examples of the Fourier transform; and sharp estimates for lower-order terms in long-time asymptotics of Ablowitz-Kaup-Newell-Segur integrable equations, with additional motivations coming from studies of perturbation of integrable equations. This mathematics research project is in the area of harmonic analysis, in which data is analyzed by breaking it into more manageable components. An important tool is the Fourier transform, which provides methods to decompose complicated signals into basic waves. These decompositions and related techniques have a number of useful applications in engineering and other applied sciences. For example, in image processing, harmonic analysis is useful in de-noising images and analyzing image structures. More generally, harmonic analysis could be used in signal processing to compress and decompress data, allowing for very large amounts of information to be effectively transmitted with manageable errors. Other variants of the Fourier transform also appear in statistics and mathematical physics. One such variant occurs in the study of energy levels of random physical systems. This research project seeks to improve the understanding of the above studies by investigating fundamental properties of the new variants of the Fourier transform.