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Title: Generalized Bernstein polynomials and total positivity
Author: Oruç, Halil
ISNI:       0000 0004 2741 4262
Awarding Body: University of St Andrews
Current Institution: University of St Andrews
Date of Award: 1999
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This thesis deals mainly with geometric properties of generalized Bernstein polynomials which replace the single Bernstein polynomial by a one-parameter family of polynomials. It also provides a triangular decomposition and 1-banded factorization of the Vandermonde matrix. We first establish the generalized Bernstein polynomials for monomials, which leads to a definition of Stirling polynomials of the second kind. These are q-analogues of Stirling numbers of the second kind. Some of the properties of the Stirling numbers are generalized to their q-analogues. We show that the generalized Bernstein polynomials are monotonic in degree n when the function ƒ is convex ... Shape preserving properties of the generalized Bernstein polynomials are studied by making use of the concept of total positivity. It is proved that monotonic and convex functions produce monotonic and convex generalized Bernstein polynomials. It is also shown that the generalized Bernstein polynomials are monotonic in the parameter q for the class of convex functions. Finally, we look into the degree elevation and degree reduction processes on the generalized Bernstein polynomials.
Supervisor: Phillips, George McArtney Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA404.5O8 ; Polynomials