delo diplomskega seminarja
Greta Stoschitzky (Author), Miran Černe (Mentor)

Abstract

Diplomska naloga opisuje skoraj periodične funkcije in njihove Fourierove vrste. Skoraj periodična funkcija je vsaka funkcija, ki jo lahko na $\mathbb{R}$ poljubno natančno enakomerno aproksimirano s končno linearno kombinacijo kosinusov ter sinusov, oziroma s končnim trigonometričnim polinomom. Izkaže se, da je vsaka skoraj periodična funkcija omejena, enakomerno zvezna ter da je vsota in produkt skoraj periodičnih funkcij zopet skoraj periodična funkcija. Te lastnosti so za periodične funkcije trivialne, medtem ko za skoraj periodične funkcije dokaz teh lastnosti ni enostaven. Podobno kot periodične funkcije imajo tudi skoraj periodične funkcije svojo posplošeno Fourierovo vrsto. Srečamo se z vprašanjem, ali so skoraj periodične funkcije enolično določene s svojo Fourierovo vrsto in če imajo lahko različne skoraj periodične funkcije isto Fourierovo vrsto. Kot pri periodičnih funkcijah je pri skoraj periodičnih funkcijah ideja dokazov konvergence za Fourierove vrste podobna, to je definirati delne Fourierove vsote in pokazati, da konvergirajo k dani funkciji.

Keywords

matematika;skoraj periodične funkcije;Fourierove vrste;delne Fourierove vsote;konvergenca Fourierovih vrst;

Data

Language: Slovenian
Year of publishing:
Typology: 2.11 - Undergraduate Thesis
Organization: UL FMF - Faculty of Mathematics and Physics
Publisher: [G. Stoschitzky]
UDC: 517.5
COBISS: 18722649 Link will open in a new window
Views: 1215
Downloads: 196
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Other data

Secondary language: English
Secondary title: Almost periodic functions
Secondary abstract: The thesis describes almost periodic functions and their Fourier series Function defined on the real line is called almost periodic, if it can be uniformly approximated with any desired degree of accuracy by a finite linear combination of sine and cosine functions, or by a finite trigonometric polynomial. Almost periodic function is bounded, uniformly continuous and the sum and the product of almost periodic functions is also an almost periodic function. These properties are trivial for periodic functions but for almost periodic functions proving these properties might be very demanding. Almost periodic functions have also their generalized Fourier series. We thus have arrived at the basic questions: can every almost periodic function be represented with Fourier series and if a generalized Fourier series completely determines an almost periodic function. As with periodic functions, the idea of proving convergence of generalized Fourier series is to form partial Fourier sums and prove that these sums converge to the original function.
Secondary keywords: mathematics;almost periodic functions;Fourier series;partial Fourier sums;convergence of Fourier series;
Type (COBISS): Final seminar paper
Study programme: 0
Embargo end date (OpenAIRE): 1970-01-01
Thesis comment: Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages: 28 str.
ID: 11222795