Borweinovi integrali

delo diplomskega seminarja

Abstract

S pomočjo funkcije sinc Zapišemo znani Dirichletov integral. Vpeljemo normo na $L^1$ prostorih, dokažemo izrek o monotoni konvergenci, Fatoujevo lemo, Lebesgueov izrek o dominirani konvergenci in nazadnje da so funkcije $C(\mathbb{R}) \cap L^1(\mathbb{R})$ v prostoru $L^1(\mathbb{R})$ goste. Dokažemo Riemann-Lebesgueovo lemo in inverzno formulo. Vpeljemo konvolucijo in dokažemo, da je asociativna in komutativna ter s pomočjo konvolucije in Fourierove transformacije izpeljemo formulo za izračun integrala z mejama $-\infty$ in $\infty$ produkta končnega števila funkcij. Izračunamo Fourierovo transformiranko funkcije sinc in s pomočjo tega izračunamo vrednosti določenih Borweinovih integralov, za druge pa poiščemo zgornjo mejo. Nato izračunamo vrednost prvega Borweinovega integrala, ki ga s prejšnjimi metodami nismo mogli. Nazadnje s pomočjo Lebesgueovega izreka o dominirani konvergenci preučimo še obnašanje vrednosti drugih.

Keywords

Borweinovi integrali;Fourierova transformacija;konvolucija;

Data

Language:	Slovenian
Year of publishing:	2024
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[J. Genc]
UDC:	517
COBISS:	208480515
Views:	37
Downloads:	17
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Borwein integrals
Secondary abstract:	Using sinc function we write the well-known Dirichlet's integral. We define the norm on $L^1$ spaces and prove the monotone convergence theorem, Fatou's lemma, Lebesgue's dominated convergence theorem and that functions $C(\mathbb{R}) \cap L^1(\mathbb{R})$ form a dense subspace of $L^1(\mathbb{R})$. We prove Riemann-Lebesgue's lemma and Fourier inversion theorem. We define convolution and prove that it is associative and commutative and using convolution and Fourier transform we derive the formula of an integral with borders $-\infty$ and $\infty$ of a product of a finite amount of functions. We calculate the Fourier transform of the function sinc and using that we calculate the values of some of the Borwein integrals and find an upper bound for others. We then calculate the value of the Borwein integral that we couldn't calculate before with the previous methods. Using the Lebesgue's dominated convergence theorem we study the behaviour of the rest of the values of Borwein integrals.
Secondary keywords:	Borwein integrals;Fourier transform;convolution;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	30 str.
ID:	25105157