Naslov (srp)

Contributions to the theory of shift-invariant spaces: doctoral dissertation

Autor

Aksentijević, Aleksandar, 1991-

Doprinosi

Aleksić, Suzana, 1979-
Pilipović, Stevan, 1950-
Teofanov, Nenad, 1968-
Stanić, Marija, 1975-
Đorđević, Dragan, 1970-
Bojović, Dejan, 1968-
Maksimović, Snježana, 1986-

Opis (eng)

Ova doktorska disertacija istraˇzuje translaciono-invarijantne potprostore Vr prostora Soboljeva Hr(Rn), pri ˇcemu je r ∈ R. Karakterizacija prostora Vr izvrˇsena je koriˇs´cenjemfunkcije opsega, operatora opsega, operatora koji komutiraju sa translacijama i talasnimfrontom. Takod¯e, izvrˇsena je karakterizacija okvira, Risove familije i Beselove familijeuz pomo´c pomenutih operatora i posebno koriste´ci Gramovu i dualnu Gramovu matricu.Istraˇzivani su odnosi izmed¯u navedenih operatora i odred¯eni uslovi pod kojima operator koji komutira sa translacijama moˇze biti s-dijagonalizabilan i moˇze se zapisati kaokonaˇcan zbir proizvoda njegovih s-sopstvenih vrednosti i odgovarju´cih projekcija. Problem dinamiˇckog uzorkovanja za prostore Vr je reˇsen i povezani su razliˇciti pristupi teorijitranslaciono-invarijantnih prostora. Elementi prostora Vr su opisani pomo´cu talasnogfronta. Na kraju, uslovi pod kojima postoji proizvod elemenata iz posmatranih prostorai uslovi kada ´ce takav proizvod pripadati nekom translaciono-invarijantnom prostoru suodred¯eni.Disertaciju ˇcini ˇsest glava. Prva glava je uvodnog karaktera. Sastoji se iz kratkog pregleda postignutih rezultata u prostoru L2(Rn), ukljuˇcuju´ci i fokus na znaˇcaj translacionoinvarijantnih prostora i drugih pojmova koji se pominju u disertaciji. U drugoj glaviizloˇzena je teorija distribucija. Glavni alat koji se koristi u disertaciji, Furijeova transformacija, predstavljena je u tre´coj glavi. Takod¯e, prostori Soboljeva Hr(Rn), r ∈ R,i prostori DL2 (Rn), D′L2 (Rn) su predstavljeni u tre´coj glavi. Cetvrta glava sadrˇzi pros- ˇtore periodiˇcnih funkcija i periodiˇcnih distribucija, neke bitne jednakosti koje se koristeu istraˇzivanju, i teoriju o talasnom frontu. Teorija okvira u Hilbertovim prostorima jeizloˇzena u petoj glavi. Na kraju, u ˇsestoj glavi su predstavljeni originalni rezultati ovedisertacije.

Opis (srp)

This doctoral dissertation investigates shift-invariant subspaces Vr of Sobolev spacesHr(Rn), where r ∈ R. Characterization of the spaces Vr was performed using rangefunctions, range operators, shift-preserving operators, and wave front. Also, characterizations of frames, Riesz families, and Bessel families were performed using the mentionedoperators and especially using Gram’s and dual Gram’s matrix. Relationships betweenthe mentioned operators were investigated, and the conditions under which the shiftpreserving operator could be s-diagonalizable and could be written as a finite sum ofproducts of its s-eigenvalues and corresponding projections were determined. The problem of dynamical sampling for spaces Vr was solved and different approaches to the theoryof shift-invariant spaces were identified. Elements of the spaces Vr were described usinga wave front. Finally, conditions under which there exists a product of elements from theobserved spaces and conditions when such a product would belong to some shift-invariantspace were determined.The dissertation consists of six chapters. The first chapter is of an introductory nature.It consists of a brief overview of the achieved results in the space L2(Rn) including thefocus on the importance of shift-invariant spaces and other concepts mentioned in dissertation. The second chapter presents the theory of distributions. The main tool usedin dissertation, the Fourier transform, is presented in the third chapter. Also, Sobolevspaces Hr(Rn), r ∈ R, and spaces DL2 (Rn), D′L2 (Rn), are presented in the third chapter.The fourth chapter discusses spaces of periodic functions and periodic distributions, someimportant equalities used in research, and the theory of wave fronts. Theory of frames inHilbert spaces is presented in the fifth chapter. Finally, the sixth chapter presents originalresults of this dissertation.

Opis (srp)

-

Jezik

srpski

Datum

2024

Licenca

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