Fourierova transformacija i Paley-Wienerov prostor

master's thesis

• Author: Antonija Kovačević
• Mentor: Damir Bakić (mentor)
• Committee member: Damir Bakić (predsjednik povjerenstva)
• Committee member: Zoran Vondraček (član povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-11-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovome radu izložene su osnove Fourierove transformacije na Banachovom prostoru \(L^1(\mathbb{R})\) i Hilbertovom prostoru \(L^2(\mathbb{R})\). Također, opisana su glavna svojstva Paley-Wienerovog prostora i dokazan je teorem o uzorkovanju na tom prostoru. Rad je podijeljen u četiri idejne cjeline, u prvom poglavlju navedeni su rezultati iz teorije normiranih prostora, te mjere i integrala koji su potrebni u proučavanju Fourierove transformacije. U drugome dijelu pažnju smo posvetili baznim okvirima koji se pokazuju korisnima u samom teoremu o uzorkovanju. Treće poglavlje posvećeno je Fourierovoj transformaciji. U četvrtom poglavlju uvodi se Paley-Wienerov prostor, proučavaju se svojstva tog prostora i na samom kraju, dokaže se teorem o uzorkovanju.

Abstract (english)

In this thesis we introduce outlines of Fourier transform in Banach space \(L^1(\mathbb{R})\) and Hilbert space \(L^2(\mathbb{R})\). Also, main properties of Paley-Wiener space is disscused and Sampling theorem has been proved. The thesis consists of four parts. Chapter 1 is a review of normed linear spaces theory and measure and integration theory, needed in coming chapters. Chapter 2 is devoted to frames, which will show useful in the Sampling theorem. In Chapter 3 we commit to Fourier transform. In final chapter, Paley-Wiener space has been introduced, main properties have been brought up and Classical Sampling Theorem has been demonstrated.

• Keywords: Fourierove transformacije; Banachov prostor; Hilbertov prostor; Paley-Wienerov prostor
• Keywords (english): Fourier transform; Banach space; Hilbert space; Paley-Wiener space
• Language: croatian
• URN:NBN: urn:nbn:hr:217:862953
• Study programme: Title: Applied 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: 2018-05-25 11:59:45