Konačni Parsevalovi bazni okviri

master's thesis

• Author: Filip Jurman
• Mentor: Ljiljana Arambašić (mentor)
• Committee member: Ljiljana Arambašić (predsjednik povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Committee member: Željka Milin Šipuš (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj rad sastoji se od četiri cjeline. U prvoj cjelini definiramo \(\mathbb{R}^n\)-bazni okvir i istražujemo njegova svojstva. Nakon toga definiramo Parsevalove skupove u \(\mathbb{R}^n\) pomoću kojih definiramo Parsevalove bazne okvire. Donosimo karakteristike koje su važne za njih te dajemo jedan primjer Parsevalovog baznog okvira iz kojeg se lijepo vide njegove specifičnosti. Na kraju cjeline govorimo o napetim baznim okvirima u prostoru \(\mathbb{R}^2\) te dajemo njihovu geometrijsku interpretaciju. U drugoj cjelini, za proizvoljne konačnodimenzionalne unitarne prostore definiramo općenite bazne okvire, čime proširujemo pojam \(\mathbb{R}^n\)-baznih okvira. U toj cjelini istražujemo rezultate i svojstava operatora analize i sinteze, kao i operator baznog okvira te Grammov operator. U trećoj cjelini bavimo se dualnim baznim okvirima, opisujemo njihova svojstva te spominjemo kanonske dualne bazne okvire. Specifično svojstvo Parsevalovih baznih okvira je da su to bazni okviri koji su sami sebi dualni, te su zbog doga najpogodniji u rekonstrukciji vektora. Na kraju rada predstavljamo numeričke algoritme pomoću kojih se mogu rekonstruirati vektori bez invertiranja operatora baznog okvira.

Abstract (english)

This thesis consists of four parts. In the first part we define the \(\mathbb{R}^n\)-frame and investigate its properties. Then we define Parseval sets in \(\mathbb{R}^n\), which we use to define Parseval frames. We present their most important properties, and give an example of a Parseval frame which shows their advantages among other frames. At the end of this chapter we consider tight frames in \(\mathbb{R}^2\) and give their geometric interpretation. In the second part we define general frames for arbitrary finitedimensional inner product spaces, which extends the term of \(\mathbb{R}^n\)-frames. In this chapter we investigate results and properties of the analysis and synthesis operators, as well as the frame operator, and Grammian operator. In the third part we examine dual frames, describe their properties, in particular the canonical dual frames. Specific property of Parseval frames is that they are frames which are dual to itself, and therefore optimal for reconstruction of vectors. Finally, we present numerical algorithms which can be used to reconstruct vectors without inverting the frame operator.

• Keywords: Parsevalovi skupovi; Parsevalovi bazni okviri; konačnodimenzionalni unitarni prostori; operatori analize i sinteze; operator baznog okvira; Grammov operator
• Keywords (english): Parseval sets; Parseval frames; finitedimensional inner product spaces; analysis and synthesis operators; frame operator; Grammian operator
• Language: croatian
• URN:NBN: urn:nbn:hr:217:095666
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-02-05 12:19:07