دانلود رایگان مقاله abscissae Greville از پایگاه های کاملا مثبت

1. Introduction

Integral recurrence formulae for B-splines have been often used in the past. The definition of B-spline as a divided difference of a truncated power function and the Hermite–Gennochi formula lead to integral recursions. One of the first papers where it is observed that the sequence of B-spline bases can be obtained by successive integration in the general context of Chebyshevian splines is Bister and Prautzsch (1997). Bernstein polynomials as well as many other examples of totally positive bases in extended Chebyshev spaces are included in this setting. Totally positive bases (TP) are bases whose collocation matrices have nonnegative minors. This kind of bases are commonly used in computer-aided design due to their shape preserving properties (see Goodman, 1996). Among all normalized TP bases of a space, we can find normalized B-bases, which are the optimal shape preserving bases (cf. Carnicer and Peña, 1994). Spaces containing algebraic polynomials and trigonometric or hyperbolic functions have attracted much interest in the field of computer-aided geometric design (Zhang, 1996; Mainar et al., 2001). In Chen and Wang (2003), integral constructions of Bernstein-like basis for cycloidal spaces Cn = cos t, sint, 1,t,...,t n−2 have been provided. In Costantini et al. (2005), such constructions are discussed in a more general setting, showing that the integral constructions provide TP bases. In particular, the normalized B-basis is expressed using integrals of a B-basis of the space of derivatives. Greville abscissae are the coefficients of the function t with respect to a given basis and play a fundamental role in the definition of Bernstein-like operators in spaces of exponential polynomials (cf. Aldaz et al., 2009).