# IJPAM: Volume 65, No. 3 (2010)

**EVALUATION OF BETA-FUNCTION B-SPLINES, II:**

LOCAL BERNSTEIN BASES

LOCAL BERNSTEIN BASES

R&D Group for Mathematical Modelling

Numerical Simulation and Computer Visualization

Faculty of Technology

Narvik University College

2, Lodve Lange's Str., P.O. Box 385, N-8505, Narvik, NORWAY

e-mail: [email protected]

url: http://ansatte.hin.no/ltd/

**Abstract.**This is the second paper in a sequence of three papers on the evaluation of *Beta-function B-splines* (BFBS), the first paper in the sequence being [#!d1-2010!#]. This sequence of papers studies explicit representations of BFBS yielding computationally efficient explicit formulae for evaluation of BFBS in terms of polynomial bases used in data interpolation, data fitting and geometric modelling, as well as in the design of multilevel constructions such as, e.g., multiwavelets. While in [#!d1-2010!#] an *interpolatory representation* of BFBS was developed in terms of *local monomial bases*, the objective of the present paper is to provide a *Bezier-type representation* in *local Bernstein bases*; the last of the three papers will be dedicated to a representation of BFBS in *global monomial bases*, suitable for use, e.g., in relevance to computing Fourier, Laplace and other transforms.

**Received: **February 12, 2010

**AMS Subject Classification: **33B15, 33B20, 41A15, 65D10, 33F05, 41A30, 65D05, 65D07, 65D20, 65D30

**Key Words and Phrases: **spline, B-spline, expo-rational, generalized,
polynomial, special function,
Gamma-function, Euler Beta-function,
complete, incomplete, Beta-function B-spline, integration,
monomial basis, Bernstein basis, Bezier representation,
local, global, interpolation, fitting, geometric modelling,
computer-aided geometric design, Fourier transform, Laplace transform

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2010

**Volume:** 65

**Issue:** 3