Parsevalovi bazni okviri Hilbertovih prostora

master's thesis

• Author: Matko Grbac
• Mentor: Damir Bakić (mentor)
• Committee member: Damir Bakić (predsjednik povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Matija Kazalicki (član povjerenstva)
• Committee member: Eduard Marušić-Paloka (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 najprije izlažemo neke osnovne pojmove i rezultate iz teorije normiranih prostora koji će se koristiti tijekom daljnjeg proučavanja. Nakon toga uvodimo pojmove Besselovog niza i baznog okvira, pojmove koji će biti središnji dio naših promatranja. Poseban naglasak imaju Parsevalovi bazni okviri. Opisuju se njihova osnovna svojstva te veze između navedenih pojmova. Poseban dio posvećen je dualnim baznim okvirima te njihovoj karakterizaciji pomoću ograničenih operatora koji posjeduju lijevi inverz. Zatim definiramo viškove baznih okvira, navodimo neka svojstva, te naposljetku pomoću istih dolazimo do karakterizacije baznih okvira koji posjeduju Parsevalov dual. Nadalje, opisujemo Besselove nizove koji dopuštaju konačno proširenje do baznog okvira. Za sam kraj rada izlažemo i kratko komentiramo fundamentalnu jednakost za Parsevalove bazne okvire.

Abstract (english)

In this thesis we first give some of the fundamental terms and results from the theory of normed spaces which will be used during further study. Afterwards, we introduce Bessel sequences and frames, terms which will be central part of our study. Special emphasis is put on Parseval frames. We describe their properties and the connections between aforementioned terms. Special part is dedicated to dual frames and their characterization using bounded operators which possess left inverse. Next we define excesses of frames, state some of their properties, and finally give a characterization of frames that possess Parseval dual by using excesses of frames. Furthermore, we describe Bessel sequences that allow finite extension to a frame. In the last part we give a fundamental identity for Parseval frames, as well as a brief discussion considering this identity.

• Keywords: teorija normiranih prostora; Besselov niz; bazni okvir; Parsevalovi bazni okviri
• Keywords (english): theory of normed spaces; Bessel sequence; frames; Parseval frames
• Language: croatian
• URN:NBN: urn:nbn:hr:217:562271
• 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: 2019-04-23 10:50:47