Müntz-Szászov teorem

master's thesis

• Author: Kristian Vedran Budrovčan
• Mentor: Tomislav Berić (mentor)
• Committee member: Tomislav Berić (predsjednik povjerenstva)
• Committee member: Ljiljana Arambašić (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Committee member: Matija Kazalicki (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Na početku rada napravljen je dokaz Weierstrassovog teorema aproksimacije koji kaže da je prostor algebarskih polinoma gust u \(C[0,1]\). Drugo poglavlje posvećeno je dokazu glavnog rezultata u radu, a to je Müntz-Szászov teorem koji karakterizira skupove potencija \(\Pi(\Lambda)=\mathrm{span}\{x^{\lambda_k}:k \in \mathbb{N}_0,\; 0=\lambda_0<\lambda_1<\cdots\}\) koji razapinju gust skup u prostoru neprekidnih funkcija na segmentu \([0,1]\). U trećem poglavlju napravljen je općenitiji slučaj teorema pod nazivom Puni Müntzov teorem. Ovdje nas također zanima gustoća skupa \(\Pi(\Lambda)\) u nekom prostoru funkcija, ali razlika u odnosu na početni teorem je što ovdje nije dan rastući niz realnih brojeva, već radimo s proizvoljnim nizovima realnih brojeva. Veći dio poglavlja zauzimaju dokazi Punog Müntzovog teorema za prostore funkcija \(C[0,1]\) i \(L^p[0,1]\), za \(0<p<\infty\). Nakon toga razmotren je Puni Müntzov teorem za \(C[a,b]\), gdje su \(0<a<b\), a na samom kraju navedeni su neki rezultati Punog Müntzovog teorema za \(C(A)\), gdje je \(A\subset [0,\infty)\) neki skup pozitivne Lebesgueove mjere.

Abstract (english)

At the beginning of this thesis, we prove the Weierstrass Approximation Theorem which says that the space of algebraic polynomials is dense in \(C[0,1]\). The second chapter is devoted to the proof of the main result in this thesis, which is the Müntz-Szász Theorem that characterizes sets of powers \(\Pi(\Lambda)=\mathrm{span}\{x^{\lambda_k}:k \in \mathbb{N}_0,\; 0=\lambda_0<\lambda_1<\cdots\}\) that span a dense subset in the space of continuous functions on the segment \([0,1]\). In the third chapter we consider a more general case of the theorem called Full Müntz Theorem. Here we are also interested in the density of the set \(\Pi(\Lambda)\) in some space of functions, but the difference from the initial theorem is that here we aren’t given an increasing sequence of real numbers, but instead we are working with arbitrary sequences of real numbers. Most of the chapter is taken up by the proof of the Full Müntz Theorem for spaces of functions \(C[0,1]\) and \(L^p[0,1]\), for \(L^p[0,1]\). After that, the Full Müntz Theorem for \(L^p[0,1]\), where \(0<a<b\), is examined, and at the very end a few results of the Full Müntz Theorem for \(C(A)\) are listed, where \(A\subset [0,\infty)\) is a set of positive Lebesgue measure.

• Keywords: Weierstrassov teorem aproksimacije; Müntz-Szászov teorem; Puni Müntzov teorem
• Keywords (english): Weierstrass Approximation Theorem; Müntz-Szász Theorem; Full Müntz Theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:828974
• Study programme: Title: Mathematical Statistics; 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: 2021-03-04 12:51:51