Gaborovi bazni okviri

master's thesis

• Author: Luka Cigler
• Mentor: Damir Bakić (mentor)
• Committee member: Damir Bakić (predsjednik povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-24, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu cilj nam je definirati i analizirati Gaborove bazne okvire na Hilbertovom prostoru $L^2(\mathbb{R})$. Da bismo to mogli napraviti, u 1. poglavlju najprije navodimo kratak pregled osnovnih rezultata iz opće teorije normiranih prostora, te definiciju i osnovna svojstva Fourierove transformacije. Zatim u 2. poglavlju definiramo bazne okvire na općenitom Hilbertovom prostoru i dokazujemo neka svojstva i karakterizacije, pri čemu su nam posebno važna rekonstrukcijska svojstva baznih... More okvira. Proučavajući rekonstrukcijska svojstva, prirodno nam se javlja pojam duala baznog okvira, pa je kraj 2. poglavlja posvećen upravo toj temi. Napokon, u 3. poglavlju definiramo Gaborove sisteme i specijalno, centralni objekt ovog rada, Gaborove bazne okvire, te koristeći do tada razvijenu teoriju proučavamo njihova svojstva. Najprije se ograničavamo na slučaj u kojem generirajuća funkcija ima kompaktan nosač, a nakon toga postepeno proširujemo razmatranja na proizvoljnu funkciju iz prostora $L^2(\mathbb{R})$. Na samom kraju rada, dokazujemo tri teorema u kojima su navedeni neki dovoljni uvjeti na generirajuću funkciju i parametre, uz koje je pripadni Gaborov sistem (Gaborov) bazni okvir za $L^2(\mathbb{R})$. Less

Abstract (english)

In this thesis our objective is to define and analyze Gabor frames on Hilbert space $L^2(\mathbb{R})$. In order to achieve this, in first chapter we list some fundamental results in general normed space theory, and we also have a brief discussion about Fourier transform. Afterwards, in the second chapter, we define frames in general Hilbert space and we prove some main properties and characterizations, most important of which are the reconstruction properties. While studying the... More reconstruction properties, the notion of dual frames naturally appears, thus, the end of the second chapter is dedicated to them. Finally, in the third chapter, we define Gabor systems, and in particular, the central objects of this thesis, the Gabor frames. Using the theory we developed earlier, we study some of their main properties. First, we limit ourselves on special case, in which the generating function of the system is compactly supported. Thereafter, we generalize these results on arbitrary functions in $L^2(\mathbb{R})$. At the very end of the thesis, we give some sufficient conditions on generating function and parameters, for which the according Gabor system is in fact a (Gabor) frame. Less

• Keywords: Gaborovni bazni okviri; Hilbertov prostor; Fourierova transformacija
• Keywords (english): Gabor frames; Hilbert space; Fourier transform
• Language: croatian
• URN:NBN: urn:nbn:hr:217:604186
• Study programme: Title: Pure Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-03-05 12:11:32