Abstract
Recently, linear positive operators of Bernstein–Schoenberg type, relative to B-splines bases, have been considered. The properties of these operators are derived mainly from the total positivity of normalized B-spline bases.
In this paper we shall construct a generalization of the operator considered in [15] by means of normalized totally positive bases generated by a particular class of totally positive scaling functions. Next, we shall study its approximation properties. Our results can be established also for more general sequences of normalized totally positive bases.
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References
J.M. Carnicer and J.M. Peña, Totally positive bases for shape preserving curves design and optimality of B-splines, Comput. Aided Geom. Design 11 (1994) 633–654.
C.K. Chui, An Introduction to Wavelets (Academic Press, San Diego, CA, 1992).
M. Cotronei and M.L. Lo Cascio, On the construction and behaviour of a general class of biorthogonal filters, in: Proceedings of AMCTM2000 (Lisbona), in press.
R.A. De Vore, Degree of approximation, in: Approximation Theory II, eds. G.G. Lorentz et al. (Academic Press, New York, 1976) pp. 117–161.
M.M. Derriennic, Sur l'approximation de fonctions intégrables sur [0,1] par les polynômes de Bernstein modifiés, J. Approx. Theory 31 (1981) 325–343.
W. Gautschi, L. Gori and F. Pitolli, Gaussian quadrature rules for refinable weights functions, Appl. Comput. Harmon. Anal. 8 (2000) 249–257.
T.N.T. Goodman, Total positivity and the shape of curves, in: Total Positivity and its Applications, eds. M. Gasca and C.A. Micchelli (Kluwer Academic Publishers, 1996) pp. 157–186.
L. Gori, L. Pezza and F. Pitolli, A class of totally positive blending B-bases, in: Curve and Surface Design: Saint Malo 1999, eds. P.-J. Laurent, P. Sablonnière and L.L. Schumaker (Vanderbilt University Press, Nashville, TN, 2000) pp. 119–126.
L. Gori, F. Pitolli and L. Pezza, On the Galerkin method based on a particular class of scaling functions, submitted.
L. Gori and F. Pitolli, Multiresolution analysis based on certain compactly supported refinable functions, in: Approximation and Optimization (Cluj-Napoca, 1996), eds. G. Coman, W.W. Breckner and P. Blaga (Transilvania Press, Cluj-Napoca, 1997) pp. 81–90.
L. Gori and F. Pitolli, A class of totally positive refinable functions, Rend. Mat. 7(20) (2000) 305–322.
L. Gori and F. Pitolli, On some applications of a class of totally positive bases, in press.
S. Karlin, Total Positivity (Stanford University Press, Stanford, 1968).
F. Pitolli, Refinement masks of Hurwitz type in the cardinal interpolation problem, Rend. Mat. 7(18) (1998) 549–563.
P. Sablonnière, Positive spline operators and orthogonal splines, J. Approx. Theory 54 (1988) 28–42.
I.J. Schoenberg, Notes on spline functions v. orthogonal or Legendre splines, J. Approx. Theory 13 (1975) 84–104.
L.L. Schumaker, Spline Functions: Basic Theory (John Wiley and Sons, New York, 1981).
G. Strang and T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge Press, Wellesley, MA, 1996).
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Gori, L., Pitolli, F. & Santi, E. Positive Refinable Operators. Numerical Algorithms 28, 199–213 (2001). https://doi.org/10.1023/A:1014003102167
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DOI: https://doi.org/10.1023/A:1014003102167