An introduction to Fourier analysis

U ovom radu proučavaju se osnove Fourierove analize: Fourierov red i Fourierova transformacija. U prvom dijelu objašnjeno je kako se periodička funkcija f : [-π, π ] → ℝ može zapisati u obliku Fourierovog rada, te navedeno je nekoliko primjera razvoja elementarnih funkcija u Fourierov red. Pokazani su bitni rezultati o konvergenciji Fourierovog reda po točkama i uniformnoj konvergenciji. U drugom dijelu rada uvedena je Fourierova transformacija na prostorima L¹ (ℝ) i L² (ℝ). Također, navedeni su primjeri Fourierove transformacije za elementarne funkcije uz neka osnovna svojstva Fourierove transformacije na prostorima L¹ (ℝ) i L² (ℝ), te je definirana i inverzna Fourierova transformacija. Na kraju rada pokazani su neke primjene Fourierove transformacije na parcijalne diferencijalne jednadžbe i na diferencijalne i integralne jednadžbe.This thesis examines the fundamentals of Fouriers analysis: Fourier series and Fourier transform. The first part explains how periodic function f : [-π, π] → ℝ can be written in the form of a Fourier series and lists several examples of Fourier series for elementary functions. It shows important results for two kinds of convergence of a Fourier series: pointwise convergence and uniform convergence. In the second part of this thesis the definition of the Fourier transform in L¹ (ℝ) and L² (ℝ) is given as well as examples of Fourier transforms for elementary functions. We also consider some basic properties of Fourier transform in L¹ (ℝ) and L² (ℝ). We also give the definition of the inverse Fourier transform. In the last part of the thesis we show some applications of the Fourier transform to partial differential equations, integral and ordinary differential equations