Processing math: 100%
prikaz prve stranice dokumenta Nelinearna Fourierova analiza sa SU (1,1) vrijednostima
  Download
PDF 891.98 KB
doctoral thesis
Nelinearna Fourierova analiza sa SU (1,1) vrijednostima
Zagreb: University of Zagreb, Faculty of Science, 2017. urn:nbn:hr:217:895781
Rupčić, Jelena
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Rupčić, J. (2017). Nelinearna Fourierova analiza sa SU (1,1) vrijednostima (Doctoral thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:895781

Rupčić, Jelena. "Nelinearna Fourierova analiza sa SU (1,1) vrijednostima." Doctoral thesis, University of Zagreb, Faculty of Science, 2017. https://urn.nsk.hr/urn:nbn:hr:217:895781

Rupčić, Jelena. "Nelinearna Fourierova analiza sa SU (1,1) vrijednostima." Doctoral thesis, University of Zagreb, Faculty of Science, 2017. https://urn.nsk.hr/urn:nbn:hr:217:895781

Rupčić, J. (2017). 'Nelinearna Fourierova analiza sa SU (1,1) vrijednostima', Doctoral thesis, University of Zagreb, Faculty of Science, accessed 29 March 2025, https://urn.nsk.hr/urn:nbn:hr:217:895781

Rupčić J. Nelinearna Fourierova analiza sa SU (1,1) vrijednostima [Doctoral thesis]. Zagreb: University of Zagreb, Faculty of Science; 2017 [cited 2025 March 29] Available at: https://urn.nsk.hr/urn:nbn:hr:217:895781

J. Rupčić, "Nelinearna Fourierova analiza sa SU (1,1) vrijednostima", Doctoral thesis, University of Zagreb, Faculty of Science, Zagreb, 2017. Available at: https://urn.nsk.hr/urn:nbn:hr:217:895781

Metadata
TitleNelinearna Fourierova analiza sa SU (1,1) vrijednostima
Title (english)Nonlinear SU(1,1)-valued Fourier analysis
AuthorJelena Rupčić
MentorVjekoslav Kovač (mentor)
Committee memberHrvoje Šikić (predsjednik povjerenstva)
Committee memberVjekoslav Kovač (član povjerenstva)
Committee memberBiserka Kolarec (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2017-06-27, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Universal decimal classification
(UDC)		51 - Mathematics
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 Lp metrike na skupu... More 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 L1 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. Less
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 Lp 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)... More 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 ρ on the group SU(1, 1) and then define a metric dp, a variant of Lp metric, on the set of all measurable SU(1, 1)-valued functions on one-dimensional torus. We characterize convergence in the metric dp 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 Lp. 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 Cp 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 L1 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 Cp=Bp, where Bp is the Babenko-Beckner constant. Less
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:895781
Study programmeTitle: Mathematics
Study programme type: university
Study level: postgraduate
Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
Type of resourceText
Extentv, 105 str.
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2019-03-21 13:09:09
accessibility

closeAccessibilityrefresh

If you wish to permanently save changes, click on Save, if not - your settings will be reset when you restart the browser.