Prilozi teoriji translaciono-invarijantnih prostora

[A. S. Aksentijević]

V, 107, [1] list

This doctoral dissertation investigates shift-invariant subspaces Vr of Sobolev spaces Hr (R n ), where r ∈ R. Characterization of the spaces Vr was performed using range functions, range operators, shift-preserving operators, and wave front. Also, characterizations of frames, Riesz families, and Bessel families were performed using the mentioned operators and especially using Gram’s and dual Gram’s matrix. Relationships between the mentioned operators were investigated, and the conditions under which the shiftpreserving operator could be s-diagonalizable and could be written as a finite sum of products of its s-eigenvalues and corresponding projections were determined. The problem of dynamical sampling for spaces Vr was solved and different approaches to the theory of shift-invariant spaces were identified. Elements of the spaces Vr were described using a wave front. Finally, conditions under which there exists a product of elements from the observed spaces and conditions when such a product would belong to some shift-invariant space 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 L 2 (R n ) including the focus on the importance of shift-invariant spaces and other concepts mentioned in dissertation. The second chapter presents the theory of distributions. The main tool used in dissertation, the Fourier transform, is presented in the third chapter. Also, Sobolev spaces Hr (R n ), r ∈ R, and spaces DL2 (R n ), D′ L2 (R n ), are presented in the third chapter. The fourth chapter discusses spaces of periodic functions and periodic distributions, some important equalities used in research, and the theory of wave fronts. Theory of frames in Hilbert spaces is presented in the fifth chapter. Finally, the sixth chapter presents original results of this dissertation.

Ova doktorska disertacija istraˇzuje translaciono-invarijantne potprostore Vr prostora Soboljeva Hr (R n ), pri ˇcemu je r ∈ R. Karakterizacija prostora Vr izvrˇsena je koriˇs´cenjem funkcije opsega, operatora opsega, operatora koji komutiraju sa translacijama i talasnim frontom. Takod¯e, izvrˇsena je karakterizacija okvira, Risove familije i Beselove familije uz 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 kao konaˇcan zbir proizvoda njegovih s-sopstvenih vrednosti i odgovarju´cih projekcija. Problem dinamiˇckog uzorkovanja za prostore Vr je reˇsen i povezani su razliˇciti pristupi teoriji translaciono-invarijantnih prostora. Elementi prostora Vr su opisani pomo´cu talasnog fronta. Na kraju, uslovi pod kojima postoji proizvod elemenata iz posmatranih prostora i uslovi kada ´ce takav proizvod pripadati nekom translaciono-invarijantnom prostoru su odred¯eni. Disertaciju ˇcini ˇsest glava. Prva glava je uvodnog karaktera. Sastoji se iz kratkog pregleda postignutih rezultata u prostoru L 2 (R n ), ukljuˇcuju´ci i fokus na znaˇcaj translacionoinvarijantnih prostora i drugih pojmova koji se pominju u disertaciji. U drugoj glavi izloˇzena je teorija distribucija. Glavni alat koji se koristi u disertaciji, Furijeova transformacija, predstavljena je u tre´coj glavi. Takod¯e, prostori Soboljeva Hr (R n ), r ∈ R, i prostori DL2 (R n ), D′ L2 (R n ) su predstavljeni u tre´coj glavi. Cetvrta glava sadrˇzi pros- ˇ tore periodiˇcnih funkcija i periodiˇcnih distribucija, neke bitne jednakosti koje se koriste u istraˇzivanju, i teoriju o talasnom frontu. Teorija okvira u Hilbertovim prostorima je izloˇzena u petoj glavi. Na kraju, u ˇsestoj glavi su predstavljeni originalni rezultati ove disertacije.

Sobolev spaces, shift-invariant space, range function, range operator, shift-preserving operator, frame, s-diagonalization, dynamical sampling, wave front, product of distributions.

Soboljevi prostori, translaciono-invarijantni prostori, funkcija opsega, operator opsega, operator koji komutira sa translacijama, okvir, s-dijagonalizacija, dinamiˇcko uzorkovanje, talasni front, proizvod distribucija.

517.98(043.3)

Matematika - Funkcionalna analiza

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