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master's thesis
Hilbertova nejednakost
Zagreb: University of Zagreb, Faculty of Science, 2022. urn:nbn:hr:217:775098
Krezo, Karla
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Krezo, K. (2022). Hilbertova nejednakost (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:775098

Krezo, Karla. "Hilbertova nejednakost." Master's thesis, University of Zagreb, Faculty of Science, 2022. https://urn.nsk.hr/urn:nbn:hr:217:775098

Krezo, Karla. "Hilbertova nejednakost." Master's thesis, University of Zagreb, Faculty of Science, 2022. https://urn.nsk.hr/urn:nbn:hr:217:775098

Krezo, K. (2022). 'Hilbertova nejednakost', Master's thesis, University of Zagreb, Faculty of Science, accessed 23 November 2024, https://urn.nsk.hr/urn:nbn:hr:217:775098

Krezo K. Hilbertova nejednakost [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2022 [cited 2024 November 23] Available at: https://urn.nsk.hr/urn:nbn:hr:217:775098

K. Krezo, "Hilbertova nejednakost", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2022. Available at: https://urn.nsk.hr/urn:nbn:hr:217:775098

Metadata
TitleHilbertova nejednakost
Title (english)Hilbert's inequality
AuthorKarla Krezo
MentorTomislav Berić (mentor)
Committee memberTomislav Berić (predsjednik povjerenstva)
Committee memberBoris Širola (član povjerenstva)
Committee memberFilip Najman (član povjerenstva)
Committee memberMarcela Hanzer (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2022-03-03, 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
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:775098
Study programmeTitle: Finance and Business Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2022-03-28 11:32:43