| Abstract | Nelinearna Fourierova analiza teorija je Fourierovih redova i Fourierove transformacije napravljena u slučaju grupe SU(1, 1), sačinjene od matrica reda 2 posebnog oblika, koji se smatra najjednostavnijim netrivijalnim slučajem. U prvom dijelu rada ispituje se konvergencija lakunarnog SU(1,1) trigonometrijskog produkta s kvadratno sumabilnim koeficijentima po odgovarajućoj metrici i g.s. U tu svrhu definira se metrika na grupi SU(1, 1), a zatim se definira i varijanta \(L^p\) metrike na skupu izmjerivih funkcija na jednodimenzionalnom torusu s vrijednostima u SU(1, 1), s obzirom na koju se proučava konvergencija. Drugi dio ovog rada proučava nelinearni analogon Hausdorff-Youngove nejednakosti. U postojećim dokazima te nejednakosti konstanta ovisi o \(1 < p < 2\). Ispituje se ponašanje konstante za funkcije koje imaju dovoljno malu \(L^1\) normu. Korištenjem profinjenja linearne Hausdorff-Youngove nejednakosti i perturbativnim tehnikama pokazuje se da nelinearna Hausdorff-Youngova nejednakost za fiksirani \(1 < p < 2\) ima manju gornju među od linearne. |
| Abstract (english) | Nonlinear Fourier analysis studies the Fourier series and the Fourier transform defined in the case of the group SU(1, 1), whose elements are particular matrices of the order two. In the first part of the thesis we begin with some basic definitions, results and \(L^p\) estimates from functional and Fourier analysis, which will be intensively used throughout the thesis. We also explain a motivation for our study, which comes from orthogonal polynomials. Then we define the lacunary SU(1, 1) trigonometric product, as a nonlinear analogue of the lacunary trigonometric series, and study two types of convergence of the product. For that purpose we first define a metric \(\rho\) on the group SU(1, 1) and then define a metric \(d_p\), a variant of \(L^p\) metric, on the set of all measurable SU(1, 1)-valued functions on one-dimensional torus. We characterize convergence in the metric \(d_p\) of lacunary SU(1, 1) trigonometric product with square summable coefficients, i.e. give a nonlinear variant of Zygmund's result on the convergence of lacunary trigonometric series in \(L^p\). Then we obtain a result on convergence a.e. of lacunary SU(1, 1) trigonometric product with square summable coefficients and its partial converse. The second part of the thesis begins with a short survey of the linear Fourier transform, followed by the definition of the nonlinear Fourier transform. Its different properties and estimates are given, as well as a motivation coming from eigenproblem for the Dirac operator and AKNS-ZS systems. We are focused on the nonlinear Hausdorff-Young inequality. In the statement of that inequality there is a constant \(C_p\) and the existing literature does not clarify how it depends on \(1 < p < 2\). We study the behavior of that constant in a particular case of functions whose \(L^1\) norm is sufficiently small. Proof of the obtained result is divided into two parts, depending on the closeness of the function to the set of Gaussians. Using the sharpened linear Hausdorff-Young inequality and perturbative techniques we show that the nonlinear Hausdorff-Young inequality, for a fixed \(1 < p < 2\), has a lower upper bound than the linear one. The result implies the nonlinear Hausdorff-Young inequality with an optimal constant \(C_p = B_p\), where Bp is the Babenko-Beckner constant. |
