IJPAM: Volume 65, No. 3 (2010)

BETA-FUNCTION B-SPLINES:
DEFINITION AND BASIC PROPERTIES

Lubomir T. Dechevsky
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.Beta-function B-splines (BFBS) were introduced by the author in 2006 and announced in [#!d-2009!#] as a particular example of smooth generalized expo-rational B-splines (GERBS) which are not true expo-rational B-splines (ERBS). The practical justification of the introduction of BFBS was that they offer a tradeoff between the good geometric-modelling properties of ERBS and the explicitness and simplicity of computation of other GERBS which are less smooth than ERBS.

The objectives of the present paper are:

  • To provide rigorous definition of BFBS.
  • To study the basic properties of BFBS and to explore the analogy of these properties to respective 'superproperties' of ERBS. The organization of this study is, therefore, similar to the study of the basic properties of ERBS in [#!d-2006!#].
  • To provide appropriate notation and numbering in relevance to the definition and properties of BFBS which can be used for fast, yet sufficiently precise, references to the respective definitions and properties in subsequent research relevant to BFBS.


Received: January 18, 2010

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

Key Words and Phrases: special function, spline, B-spline, expo-rational, generalized, Bernstein polynomial, cumulative distribution function, density, incomplete Euler Beta-function, discontinuous, continuous, absolutely continuous, smooth, infinitely smooth, geometric modelling

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 65
Issue: 3