Dahmen, Wolfgang; Micchelli, Charles A. On multivariate E-splines. (English) Zbl 0706.41010 Adv. Math. 76, No. 1, 33-93 (1989). The paper settles some of the open problems suggested by A. Ron [Constructive Approximation 4, No.4, 357-378 (1988; Zbl 0674.41005)] and encompasses a more general algebraic study of cardinal E-splines. The authors’ approach is based on some new bounds and formulas for the dimension of certain intersections of null spaces of commuting linear operators. In particular, this achieves a characterization of the local linear independence of translates of exponential cube spline as also enables them to study local algebraic properties of functions spanned by them. The paper includes analysis of some algorithms based on the concept of discrete exponential splines. The combinatorial interpretation of the discrete splines and relationship to their continuous counterparts leads to interesting combinatorial applications for enumerating the number of solutions to linear diophantine systems. Reviewer: A.Sahai Cited in 1 ReviewCited in 28 Documents MSC: 41A15 Spline approximation Keywords:cardinal E-splines; exponential cube spline; algorithms; linear diophantine systems Citations:Zbl 0674.41005 PDFBibTeX XMLCite \textit{W. Dahmen} and \textit{C. A. Micchelli}, Adv. Math. 76, No. 1, 33--93 (1989; Zbl 0706.41010) Full Text: DOI References: [1] Dahmen, W., On multivariate \(B\)-splines, SIAM J. Numer. Anal., 17, 179-191 (1980) · Zbl 0425.41015 [2] Dahmen, W.; Micchelli, C. A., On the limits of multivariate \(B\)-splines, J. Analyse Math., 39, 256-278 (1981) · Zbl 0473.41007 [3] Dahmen, W.; Micchelli, C. A., Recent progress in multivariate splines, (Chui, C. K.; Schumaker, L. L.; Ward, J., Approximation Theory IV (1983), Academic Press: Academic Press New York), 27-121 · Zbl 0559.41011 [4] Dahmen, W.; Micchelli, C. A., Translates of multivariate splines, Linear Algebra Appl., 52/53, 217-234 (1983) · Zbl 0522.41009 [5] Dahmen, W.; Micchelli, C. A., Subdivision algorithms for the generation of box spline surfaces, Comput. Aided Geom. Design, 1, 115-129 (1984) · Zbl 0581.65011 [6] Dahmen, W.; Micchelli, C. A., On the local linear independence of translates of a box spline, Stud. Math., 82, 243-263 (1985) · Zbl 0545.41018 [7] Dahmen, W.; Micchelli, C. A., On the solution of certain systems of partial difference equations and linear dependence of translates of box splines, Trans. Amer. Math. Soc., 292, 305-320 (1985) · Zbl 0637.41012 [8] Dahmen, W.; Micchelli, C. A., Algebraic properties of discrete box splines, Constr. Approx., 3, 209-221 (1987) · Zbl 0633.41010 [9] Dahmen, W.; Micchelli, C. A., The number of solutions to linear diophantine equations and multivariate splines, Trans. Amer. Math. Soc., 308, 509-532 (1988) · Zbl 0655.10013 [10] Dahmen, W.; Micchelli, C. A., On the theory and applications of exponential splines, (Chui, C. K.; Schumaker, L. L.; Utreras, F. I., Topics in Multivariate Approximation (1989), Academic Press: Academic Press New York), 37-46 [11] Jia, R. Q., A dual basis for the integer translates of an exponential box spline (1989), preprint · Zbl 0693.41013 [12] Karlin, S.; Micchelli, C. A.; Rinott, Y., Multivariate splines: A probabilistic perspective, J. Multivariate Anal., 20, 69-90 (1986) · Zbl 0623.41014 [13] Micchelli, C. A., Cardinal \(L\)-splines, (Karlin, S.; Micchelli, C. A.; Pinkus, A.; Schoenberg, I. J., Studies in Splines and Approximation Theory (1975), Academic Press: Academic Press New York), 203-250 [14] Constr. Approx., 4, 357-378 (1988) · Zbl 0674.41005 [15] Stanley, R., Combinatorial reciprocity theorems, Adv. in Math., 14, 194-253 (1974) · Zbl 0294.05006 [16] Welsh, D. J.A, Matroid Theory (1976), Academic Press: Academic Press London/New York/San Francisco · Zbl 0343.05002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.