Dualne bazne funkcije za Bernsteinove polinome

delo diplomskega seminarja

Eva Rozman (Author), Jan Grošelj (Mentor)

Abstract

V delu je predstavljena Bernsteinova baza vektorskega prostora polinomov stopnje manjše ali enake $n$ in njene najpomembnejše lastnosti. Na kratko so opisane Bézierove krivulje in njihova uporaba v računalniško podprtem geometrijskem oblikovanju in de Casteljaujev algoritem kot stabilna metoda za iskanje vrednosti polinoma v dani točki. Izpeljana je eksplicitna formula za dualne bazne funkcije, predstavljene v obliki linearne kombinacije Bernsteinovih baznih polinomov, ter podan matrični zapis relacije med tema dvema bazama. Z dualnimi baznimi funkcijami vpeljemo dualne funkcionale, ki razpenjajo dualni vektorski prostor. Obravnavan je vektorski prostor polinomov stopnje manjše ali enake $n$ z ničelnimi robnimi pogoji in njemu prirejena Bernsteinova in dualna Bernsteinova baza. Na koncu navedemo še praktično uporabo dobljenih rezultatov, zvezno aproksimacijo po metodi najmanjših kvadratov, kjer za dano funkcijo $f$ iščemo tak polinom $p^*,$ ki minimizira drugo normo $\norm{f-p}.$

Keywords

Bernsteinova baza;dualna baza;dualni funkcionali;polinomska aproksimacija;

Data

Language:	Slovenian
Year of publishing:	2020
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[E. Rozman]
UDC:	517.9
COBISS:	58724867
Views:	741
Downloads:	89
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Dual basis functions of Bernstein polynomials
Secondary abstract:	In the paper the Bernstein basis of the vector space of polynomials of degree at most $n$ and its key properties are presented. The Bézier curves and their use in computer aided geometric design and the de Casteljau's algorithm as a stable method for finding value of polynomial in a given point are briefly discussed. An explicit formula for the dual basis functions expressed as linear combinations of Bernstein polynomials is derived and a matrix form of the relation between these two bases is given. With the Bernstein basis functions we introduce dual functionals which span the dual vector space. The vector space of polynomials of degree at most $n$ with boundary constraints is defined and both the Bernstein and the dual Bernstein basis of such space are derived. Finally, we provide a practical application of the results, the continuous least squares approximation, where, for a given function $f$, we search the polynomial $p^*$ that minimizes the second norm $\norm{f-p}.$
Secondary keywords:	Bernstein basis;dual basis;dual functionals;polynomial approximation;
Type (COBISS):	Final seminar paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Finančna matematika - 1. stopnja
Pages:	29 str.
ID:	12074669