Bazni okviri konačnodimenzionalnih Hilbertovih prostora

Abstract

U ovom radu smo se upoznali s baznim okvirima konačnodimenzionalnih Hilbertovih prostora, proučili smo njihova osnovna svojstva te istražili neke primjere. U prvom poglavlju navedene su definicije osnovnih pojmova linearne algebre kao i iskazi tvrdnji koje su nam potrebne za središnji dio rada. Na kraju poglavlja iskazan je i dokazan Parsevalov identitet. Drugo poglavlje počinje definicijom temeljnog pojma ovog diplomskog rada, baznog okvira za konačnodimenzionalne prostore, nakon čega slijede razmatranja svojstava posebnih klasa baznih okvira kao što su bazni okviri konstruirani iz zadane ortonormirane baze, Parsevalovi i napeti bazni okviri. Potom su definirani linearni operatori pridruženi baznim okvirima: operator analize, sinteze i operator baznog okvira. Također su promatrana svojstva navedenih linearnih operatora i predstavljene su karakterizacije klasa baznih okvira pomoću tih operatora. Treća sekcija posvećena je sličnim i unitarno izomorfnim baznim okvirima i njihovim karakterizacijama koristeći pripadne operatore analize i sinteze. Treće poglavlje posvećeno je baznim okvirima prostora $\mathbb{R}^2$. Nakon geometrijske interpretacije napetih baznih okvira tog prostora predstavljena je PRR-ekvivalencija te su navedene konstrukcije koje osiguravaju napetost baznog okvira u $\mathbb{R}^2$. U posljednjem poglavlju predstavljeno je jedno od najvažnijih svojstava baznih okvira, mogućnost rekonstrukcije svakog vektora prostora koristeći vektore baznog okvira, odnosno navedena je rekonstrukcijska formula te algoritmi rekonstrukcije. Za kraj poglavlja proučavani su kanonski i alternativni dualni bazni okviri i njihova svojstva.In this thesis, we have introduced the frames of the finite-dimensional Hilbert spaces, discussed their basic properties and explored some examples. The first chapter contained definitions of the basic concepts of linear algebra as well as the statements which were needed for the main part of the thesis. At the end of the chapter Parseval identity was proven. The second chapter began with the definition of the basic term of this graduate thesis, the frame for finite-dimensional spaces, followed by the consideration of the properties of the special classes of frames, such as frames constructed from given orthonormal basis, the Parseval and the tight frames. Then linear operators associated with the frames were defined: the analysis operator, the synthesis operator and the frame operator. Also, properties of these linear operators were observed and the characterizations of the certain classes of frames in the terms of these operators were presented. The third section was devoted to similar and unitary isomorphic frames and their characterizations using related analysis and synthesis operators. The third chapter was devoted to frames in $\mathbb{R}^2$. After the geometric interpretation of the tight frames of this space, PRR-equivalence was presented and replacements of vectors in a frame in order to get a tight frame for $\mathbb{R}^2$. In the last chapter the most important property of frames was presented, the possibility of reconstructing each vector in the space using frame vectors, in other words the reconstruction formula and reconstruction algorithms. At the end of the chapter, canonical and alternative dual frames and their properties were observed