Fourierova analiza na lokalno kompaktnim abelovim grupama i neke primjene

master's thesis

• Author: Ljudevit Palle
• Mentor: Vjekoslav Kovač (mentor)
• Committee member: Vjekoslav Kovač (predsjednik povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Ante Mimica (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-07-13, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom je radu dan pregled glavnih rezultata iz Fourierove analize na lokalno kompaktnim, Abelovim grupama. Na početku je iznesen osnovni materijal iz teorije mjere, topologije i funkcionalne analize potreban za razvoj teorije na lokalno kompaktnim grupama. Potom se proučavaju Geljfandova transformacija, funkcije i konvolucija na lokalno kompaktnim grupama, reprezentacije lokalno kompaktnih grupa i funkcije pozitivnog tipa. Izneseni su i dokazani Geljfand-Raikovljev teorem, teorem o Fourierovoj inverziji i teorem o Pontrjaginovoj dualnosti, a na kraju rada se jos proučava Bohrova kompaktifikacija i primjenjuje na karakterizaciju uniformno skoro periodičnih funkcija.

Abstract (english)

This work provides an overview of the more important results in Fourier analysis on locally compact Abelian groups. At the beginning the basic material from measure theory, topology, and functional analysis is listed, as it is necessary for developing the theory on locally compact groups. Afterwards, the Gelfand transform is studied, followed by functions and convolution on locally compact groups, representations of locally compact groups, and functions of positive type. The main theorems formulated and proven in this work are: the Gelfand-Raikov theorem, the Fourier inversion theorem, and the Pontrjagin duality theorem. Finally, the Bohr compactification is investigated and applied to a characterization of uniformly almost periodic functions.

• Keywords: Fourierove analiza; lokalno kompaktne Abelove grupe; Geljfandova transformacija; Geljfand-Raikovljev teorem; teorem o Fourierovoj inverziji; teorem o Pontrjaginovoj dualnosti; Bohrova kompaktifikacija
• Keywords (english): Fourier analysis; locally compact Abelian groups; Gelfand transform; Gelfand-Raikov theorem; Fourier inversion theorem; Pontrjagin duality theorem; Bohr compactification
• Language: croatian
• URN:NBN: urn:nbn:hr:217:620119
• 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: 2019-01-29 09:13:27