On convolutions of B-splines

Abstract A smooth approximation to a function ƒ is achieved by convolving ƒ with a smooth function φ. When φ is nonnegative, of unit mean value, compactly supported and has certain symmetry properties, convolving with φ respects the shape properties of the data ƒ such as local positivity, monotonicity and convexity. We study the convolution of ƒ and φ when φ is a univariate B-spline, tensor product B-spline, box spline or simplex spline, and ƒ is a linear combination of the same kind of splines as φ. In terms of divided differences and blossoms, we express the convolution of univariate splines over nonuniform knots as linear combinations of B-splines. This conversion can be carried out by a stable recurrence.

[1]  C. Micchelli,et al.  Recent Progress in multivariate splines , 1983 .

[2]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[3]  T. Lyche,et al.  Box Splines and Applications , 1991 .

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

[5]  L. Schumaker.  Spline Functions: Basic Theory , 1981 .

[6]  Gerald B. Folland,et al.  Real Analysis: Modern Techniques and Their Applications , 1984 .

[7]  Tom Lyche,et al.  Cones and recurrence relations for simplex splines , 1987 .

[8]  G. Nürnberger.  Approximation by Spline Functions , 1989 .

[9]  C. D. Boor,et al.  On Calculating with B-Splines II. Integration , 1976 .

[10]  C. D. Boor,et al.  Spline approximation by quasiinterpolants , 1973 .

[11]  Wolfgang Dahmen,et al.  Computation of inner products of multivariate B-splines , 1981 .

[12]  H. Seidel.  Representing piecewise polynomials as linear combinations of multivariate B-splines , 1992 .

[13]  Jules Bloomenthal,et al.  Convolution surfaces , 1991, SIGGRAPH.

[14]  P. Sablonnière.  Positive spline operators and orthogonal splines , 1988 .

[15]  K. Mørken.  Some identities for products and degree raising of splines , 1991 .

[16]  Klaus Höllig,et al.  B-splines from parallelepipeds , 1982 .

[17]  C. D. Boor,et al.  Splines as linear combinations of B-splines. A Survey , 1976 .

[18]  Klaus Höllig,et al.  Recurrence Relations for Multivariate B-Splines. , 1982 .

[19]  C. Micchelli.  On a numerically efficient method for computing multivariate B-splines , 1979 .

[20]  L. Schumaker,et al.  Degree raising for splines , 1986 .

[21]  R. DeVore.  The Approximation of Continuous Functions by Positive Linear Operators , 1972 .