20.500.12556/RUL-121513 Dualne bazne funkcije za Bernsteinove polinome delo diplomskega seminarja Dual basis functions of Bernstein polynomials 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 $\norm{f-p}.$ 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 $\norm{f-p}.$ Bernsteinova baza dualna baza dualni funkcionali polinomska aproksimacija Bernstein basis dual basis dual functionals polynomial approximation true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2020-10-13 11:45:23 2020-10-13 11:45:27 2024-05-29 14:27:30 0000-00-00 00:00:00 2020 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 517.9 113404 58724867 1521.pdf 1521.pdf 1 98E1F175C7AF418CB3DB1248EF5F3BFE d25e16de3410c61d10e524c94edc1044c6201dbcb1858634d6a6e5d970a59f29 e6e1ddf0-a1ba-11eb-a523-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=136448 Fakulteta za matematiko in fiziko 0 0 0