Abstract
An important aspect in the study of multivariate box splines is the fact that the set of polynomials contained in the linear span of the lattice translates of a box spline is precisely the common kernel of a family of linear partial differential operators. Dahmen and Micchelli were the first to compute the dimension of this kernel and to realize that the dimension theory extends to general linear operators indexed by a finite set X with a matroid structure. The kernel space is induced by the matroid structure on X and the dimension is given as a sum of the dimensions of simpler kernels which are also identified by the matroid structure. This paper surveys the recent developments in this area.
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References
Ben-Artzi, A. and A. Ron, Translates of exponential box splines and their related spaces, Trans. Amer. Math. Soc. 309 (1988), 683–709.
de Boor, C. and K. Höllig, B-splines from parallelepipeds, J. Analyse Math. 42 (1982/3), 99–115.
de Boor, C. and A. Ron, On multivariate polynomial interpolation, CMS TSR 89–17, University of Wisconsin, Madison, to appear in Constr. Approx.
de Boor, C. and A. Ron, On polynomial ideals of finite co-dimension with applications to box spline theory, CMS Report # 89–21, University of Wisconsin-Madison.
de Boor, C. and A. Ron, Polynomial ideals and multivariate splines, in Multivariate Approximation Theory V, W. Schempp and K. Zeller (Eds.), Birkhäuser-Verlag, Basel, 1990.
Dahmen, W., R.Q. Jia, and C.A. Micchelli, On linear dependence relations for integer translates of compactly supported distributions, to appear in Math. Nachr.
Dahmen, W. and C.A. Micchelli, Translates of multivariate splines, Linear Algebra Appl. 52/53 (1983), 217–234.
Dahmen, W. and C.A. Micchelli, On the local linear independence of translates of a box spline, Studia Math. 82 (1985), 243–263.
Dahmen, W. and C.A. Micchelli, On the solution of certain systems of partial difference equations and linear dependence of translates of box splines, Trans. Amer. Math. Soc. 292 (1985), 305–320.
Dahmen, W. and C.A. Micchelli, On multivariate E-splines, Adv. in Math. 76 (1989), 33–93.
Dahmen, W. and C.A. Micchelli, Local dimension of piecewise polynomial spaces, syzygies, and solutions of systems of partial equations, I.B.M. Research Report 1988, to appear in Math. Nachr.
Jia, R.-Q., S. Riemenschneider, and Z. Shen, Dimensions of kernels of linear operators, preprint.
Ron, A., Exponential box splines, Constr. Approx. 4 (1988), 357–378.
Shafarevich, I. R., Basic Algebraic Geometry, Springer-Verlag, New York, 1974.
Shen, Z., Dimension of certain kernel spaces of linear operators, to appear in Proc. Amer. Math. Soc.
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Riemenschneider, S.D., Jia, RQ., Shen, Z. (1990). Multivariate Splines and Dimensions of Kernels of Linear Operators. In: Haußmann, W., Jetter, K. (eds) Multivariate Approximation and Interpolation. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 94. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5685-0_20
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DOI: https://doi.org/10.1007/978-3-0348-5685-0_20
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