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Mathematical Analysis

Generalized Bernstein polynomials and total positivity

Profile image of Halil OrucHalil Oruc

1999

Abstract

This thesis submitted for Ph.D. degree deals mainly with geometric properties of generalized Bernstein polynomials which replace the single Bernstein polynomial Bn! by a one-parameter family of polynomials B~!. It also provides a triangular decomposition and l-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, that is B~! ~ B~+d, when the function! is convex. There is also a representation of B~! B~+1!involving second order divided differences of the function j'. Shape preserving properties of the generalized Bernstein polynomials are studied by making use of the concept of total positivity. It is proved that monotonic ...

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Related topics

  • Mathematics
  • Bernstein Polynomial
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