Funkcijske jednadžbe i ortogonalnost

master's thesis

• Title (english): Functional equations and orthogonality
• Author: Rahela Boro
• Mentor: Dijana Ilišević (mentor)
• Committee member: Dijana Ilišević (predsjednik povjerenstva)
• Committee member: Mea Bombardelli (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-12-01, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu definirane su Birkho–Jamesova, jednakokračna, Robertsova, Pitagorina i poluskalarna ortogonalnost. Svaka od njih poopćenje je standardne ortogonalnosti u unitarnim prostorima. Proučavana su četiri tipa uvjetnih funkcijskih jednadžbi (tzv. ortogonalnih funkcijskih jednadžbi) dobivenih dodavanjem uvjeta ortogonalnosti poznatim funkcijskim jednadžbama:

(i) aditivna: \(f(x+y) \)=\(f(x) \)+ \(f(y) \), \(\: x\perp y\),
(ii) Jensenova: \(2f\left( \tfrac{x+y}{2}\right) \) =\(\tfrac{f(x)+f(y)}{2}\), \(\: x\perp y\),
(iii) kvadratna: \(f(x+y) \)+ \(f(x-y) \)=2\(f(x) \)+2\(f(y) \), \(\: x\perp y\),
(iv) d'Alembertova: \(f(x+y) \)+ \(f(x-y) \)=2\(f(x) \)\(f(y) \), \(\: x\perp y\).

Opisano je i na koji način funkcija čuva standardnu ortogonalnost te koje uvjete mora zadovoljavati kako bi očuvala Birkho–Jamesovu, jednakokračnu i poluskalarnu ortogonalnost u normiranim prostorima.

Abstract (english)

In this thesis, Birkho–James, isosceles, Roberts, Pythagorean and semi–inner product orthogonality are defined. Each of them is a generalization of standard orthogonality in inner product spaces. Four types of conditional functional equations (so-called orthogonal functional equations) obtained by adding an orthogonality condition to known functional equations are studied:
(i) additive: \(f(x+y) \)=\(f(x) \)+ \(f(y) \), \(\: x\perp y\),
(ii) Jensen: \(2f\left( \tfrac{x+y}{2}\right) \) =\(\tfrac{f(x)+f(y)}{2}\), \(\: x\perp y\),
(iii) quadratic: \(f(x+y) \)+ \(f(x-y) \)=2\(f(x) \)+2\(f(y) \), \(\: x\perp y\),
(iv) d’Alembert: \(f(x+y) \)+ \(f(x-y) \)=2\(f(x) \)\(f(y) \), \(\: x\perp y\).
It is also described how a function preserves standard orthogonality and which conditions it must satisfy in order to preserve Birkho –James, isosceles and semi–inner product orthogonality in normed spaces.

• Keywords: Birkho–Jamesova ortogonalnost; jednakokračna ortogonalnost; Robertsova ortogonalnost; Pitagorina ortogonalnost; poluskalarna ortogonalnost
• Keywords (english): Birkho-James orthogonality; isosceles orthogonality; Roberts orthogonality; Pythagorean orthogonality; semi-scalar orthogonality
• Language: croatian
• URN:NBN: urn:nbn:hr:217:407847
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-12-14 11:30:10