Chebyshev splines beyond total positivity

For polynomial splines as well as for Chebyshev splines, it is known that total positivity of the connection matrices is sufficient to obtain B-spline bases. In this paper we give a necessary and sufficient condition for the existence of B-bases (or, equivalently, of blossoms) for splines with connection matrices and with sections in different four-dimensional extended Chebyshev spaces.

[1]  Phillip J. Barry,et al.  de Boor-Fix dual functionals and algorithms for Tchebycheffian B-spline curves , 1996 .

[2]  Marie-Laurence Mazure,et al.  Blossoming: A Geometrical Approach , 1999 .

[3]  Marie-Laurence Mazure.  Chebyshev spaces with polynomial blossoms , 1999, Adv. Comput. Math..

[4]  Tom Lyche,et al.  A recurrence relation for chebyshevianB-splines , 1985 .

[5]  Nira Dyn,et al.  ON LOCALLY SUPPORTED BASIS FUNCTIONS FOR THE REPRESENTATION OF GEOMETRICALLY CONTINUOUS CURVES , 1987 .

[6]  H.-P. Seidel.  New algorithms and techniques for computing with geometrically continuous spline curves of arbitrary degree , 1992 .

[7]  T. Goodman.  Properties of ?-splines , 1985 .

[8]  Lyle Ramshaw,et al.  Blossoms are polar forms , 1989, Comput. Aided Geom. Des..

[9]  Marie-Laurence Mazure,et al.  Polynomial Chebyshev splines , 1999, Comput. Aided Geom. Des..

[10]  Marie-Laurence Mazure,et al.  Vandermonde type determinants and blossoming , 1998, Adv. Comput. Math..

[11]  B. Barsky.  The beta-spline: a local representation based on shape parameters and fundamental geometric measures , 1981 .

[12]  Nira Dyn,et al.  Recurrence relations for Tchebycheffian B-splines , 1988 .

[13]  Tom Lyche,et al.  Construction of Exponential Tension B-splines of Arbitrary Order , 1991, Curves and Surfaces.

[14]  Helmut Pottmann,et al.  Helix splines as an example of affine Tchebycheffian splines , 1994, Adv. Comput. Math..

[15]  Helmut Pottmann,et al.  The geometry of Tchebycheffian splines , 1993, Comput. Aided Geom. Des..

[16]  Nira Dyn,et al.  Piecewise polynomial spaces and geometric continuity of curves , 1989 .

[17]  Marie-Laurence Mazure,et al.  Piecewise Smooth Spaces in Duality , 1999 .