Hilbertova nejednakost

master's thesis

• Title (english): Hilbert's inequality
• Author: Karla Krezo
• Mentor: Tomislav Berić (mentor)
• Committee member: Tomislav Berić (predsjednik povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-03-03, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu smo izveli Hilbertovu nejednakost te pokazali neke njene primjene. U prvome dijelu rada smo izveli Hilbertovu nejednakost u diskretnom i integralnom obliku te odredili da je optimalna konstanta u oba slučaja jednaka π. Za nejednakost u diskretnom obliku smo ponudili i alternativni dokaz pomoću geometrijske interpretacije. Također, izveli smo poopćenu Hilbertovu nejednakost koja vrijedi za konjugirane potencije p, q > 1 te pokazali da, uz različitu optimalnu konstantu, mogu vrijediti i slične nejednakosti, npr. varijanta s maksimumom koja nastaje zamjenom izraza m + n u nazivniku s max (m, n). Na kraju smo pokazali da je lijeva strana Hilbertove nejednakosti nenegativna. U drugome dijelu smo se bavili primjenama Hilbertove nejednakosti. Za primjene su nam najbitnije Hilbertove matrice pa smo im, nakon što smo ih definirali, iskazali i neka od njihovih važnijih svojstava, poput (anti)simetričnosti i regularnosti. Zatim smo uveli trigonometrijske polinome pomoću kojih smo dokazali još jedan oblik Hilbertove nejednakosti te odredili normu Hilbertovih matrica. Na kraju smo Hilbertovu nejednakost primijenili na integrale polinoma.

Abstract (english)

In this thesis, we derive Hilbert’s inequality and present some of its applications. In the first chapter, we derive Hilbert’s inequality in its discrete and integral forms and determine that in both cases, π is the optimal constant. Additionally, we offer an alternative proof of discrete Hilbert’s inequality using a geometric interpretation. We also derive a generalization of the discrete Hilbert’s inequality for conjugate exponents p, q > 1 and show that similar inequalities, such as the maximum variant where the expression m + n in the denominator on the left side is replaced with max (m, n), also hold. In the end, we prove that the left-hand side of the discrete Hilbert’s inequality is non-negative. In the second chapter, after once again reviewing the key notions, we define Hilbert matrices and state some of their properties, such as (skew)symmetry and regularity. Next, we introduce trigonometric polynomials and use them to prove another form of Hilbert’s inequality, as well as determine the norm of Hilbert matrices. In the end, we apply Hilbert’s inequality to integrals of polynomials.

• Keywords: optimalna konstanta; poopćenja Hilbertove nejednakosti; Hilbertove matrice; trigonometrijski polinomi
• Keywords (english): optimal constant; generalizations of Hilbert inequality; Hilbert matrices; trigonometric polynomials
• Language: croatian
• URN:NBN: urn:nbn:hr:217:775098
• Study programme: Title: Finance and Business 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: 2022-03-28 11:32:43