COURSE GOALS:
The main goal of the course is introducing students to one nonlinear model of the Fourier analysis, which finds its applications in various mathematical branches: from the theory of orthogonal polynomials to the mathematical physics. Material covered by this course corresponds to the content of the lecture notes [1], which will be eventually turned into a book [3]; also see the doctoral dissertation [6]. Some material will be complemented with small parts of the books [4] and [7],[8].
COURSE DESCRIPTION:
Fourier analysis is an old branch of the mathematical analysis, which finds its basic ideas in the work of J.-B. J. Fourier from the beginning of the 19th century. The nonlinear Fourier analysis is a term coined by T. Tao and C. Thiele [1] for one mathematical model, which has been repeatedly rediscovered throughout the history. Comparison with the Fourier analysis is natural because many results and questions regarding that model can be interpreted as nonlinear variants of the corresponding results from the Fourier analysis. A very brief introduction to the topic can be found in short expository papers [2],[5]. We will decide to primarily consider a discrete model, because it is easier to formulate, which will make the course more elementary and easier for understanding.
COURSE CONTENTS:
1. Basic theory of the (linear) Fourier series. (10 hours)
2. Definition of the nonlinear Fourier transform/series on l1. Extensions to lp, 1
3. The nonlinear Fourier transform/series on l2(N0) and l2(Z). The nonlinear Plancherel (i.e., Parseval) identity and its generalizations. Existence and uniqueness of the inverse transform. (10 hours)
4. The Riemann-Hilbert problem. (10 hours)
5. Orthogonal polynomials on the unit circle. Several other applications of the model (integrable systems, Gaussian processes). (20 hours)
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