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

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

LOCAL MONOMIAL BASES

LOCAL MONOMIAL 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: ltd@hin.no

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

**Abstract.**This is the first one of a sequence of three papers addressing the evaluation of *Beta function B-splines* (BFBS)
earlier introduced and studied in [#!d-2008!#,#!dbl-2009!#,#!d-2010!#]. BFBS are a special case of *generalized expo-rational B-splines* (GERBS) [#!d-2008!#,#!dbl-2009!#], and are a practically important instance of smooth GERBS which are not infinitely smooth true *expo-rational B-splines* (ERBS) [#!d-2004!#,#!dlb-2006!#]. Compared to ERBS, BFBS exhibit similar properties, cf. [#!dlb-2006!#,#!d-2010!#], but with more limited range, due to the polynomial nature of BFBS compared to the exponential nature of ERBS. On the other hand, the integrals in the definition of BFBS can be solved exactly in elementary functions, and the resulting representation is computationally efficient, while with ERBS the respective integrals are special functions which can be computed efficiently, yet approximately. This makes BFBS more applicable than ERBS, e.g., in topics related to refinement, subdivision, multiresolution and other multilevel techniques.

The present sequence of three papers is dedicated to the derivation of 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. This is the first paper of the sequence, and here we derive a representation of BFBS in terms of local monomial bases. This is an essentially interpolatory representation; in the second article of the sequence, a Bezier type representation of BFBS will be considered, which is suitable for geometric modelling and data fitting; the third and last paper of the sequence will be dedicated to a representation in global monomial bases, suitable for use, e.g., in relevance to certain operational calculi.

**Received: **January 28, 2010

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

**Key Words and Phrases: **spline, B-spline, exponential, rational, expo-rational,
generalized, polynomial, special function, Gamma-function, Euler Beta-function,
complete, incomplete, Beta-function B-spline, monomial basis, Bernstein basis,
local, global, interpolation, fitting, geometric modelling, operational calculus

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

**ISSN:** 1311-8080

**Year:** 2010

**Volume:** 65

**Issue:** 3