Multivariate discrete splines and linear Diophantine equations

Jia, Rong Qing

doi:10.1090/S0002-9947-1993-1159194-6

Multivariate discrete splines and linear Diophantine equations

Author: Rong Qing Jia
Journal: Trans. Amer. Math. Soc. 340 (1993), 179-198
MSC: Primary 41A15; Secondary 11D04, 39A10, 39A70, 41A63
DOI: https://doi.org/10.1090/S0002-9947-1993-1159194-6
MathSciNet review: 1159194
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Abstract: In this paper we investigate the algebraic properties of multivariate discrete splines. It turns out that multivariate discrete splines are closely related to linear diophantine equations. In particular, we use a solvability condition for a system of linear diophantine equations to obtain a necessary and sufficient condition for the integer translates of a discrete box spline to be linearly independent. In order to understand the local structure of discrete splines we develop a general theory for certain systems of linear partial difference equations. Using this theory we prove that the integer translates of a discrete box spline are locally linearly independent if and only if they are linearly independent.

Asher Ben-Artzi and Amos Ron, Translates of exponential box splines and their related spaces, Trans. Amer. Math. Soc. 309 (1988), no. 2, 683–710. MR 961608, DOI https://doi.org/10.1090/S0002-9947-1988-0961608-7
C. de Boor and R. DeVore, Approximation by smooth multivariate splines, Trans. Amer. Math. Soc. 276 (1983), no. 2, 775–788. MR 688977, DOI https://doi.org/10.1090/S0002-9947-1983-0688977-5
C. de Boor and K. Höllig, $B$-splines from parallelepipeds, J. Analyse Math. 42 (1982/83), 99–115. MR 729403, DOI https://doi.org/10.1007/BF02786872

E. Cohen, T. Lyche, and R. Riesenfeld, Discrete box splines and refinement algorithms, Computer Aided Geometric Design 1 (1984), 131-148.

Wolfgang Dahmen, On multivariate $B$-splines, SIAM J. Numer. Anal. 17 (1980), no. 2, 179–191. MR 567267, DOI https://doi.org/10.1137/0717017
Wolfgang Dahmen and Charles A. Micchelli, Recent progress in multivariate splines, Approximation theory, IV (College Station, Tex., 1983) Academic Press, New York, 1983, pp. 27–121. MR 754343
Wolfgang Dahmen and Charles A. Micchelli, Translates of multivariate splines, Linear Algebra Appl. 52/53 (1983), 217–234. MR 709352, DOI https://doi.org/10.1016/0024-3795%2883%2980015-9

---, Subdivision algorithms for the generation of box spline surfaces, Computer Aided Geometric Design 1 (1984), 115-129.

Wolfgang Dahmen and Charles A. Micchelli, On the local linear independence of translates of a box spline, Studia Math. 82 (1985), no. 3, 243–263. MR 825481, DOI https://doi.org/10.4064/sm-82-3-243-263
Wolfgang Dahmen and Charles 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), no. 1, 305–320. MR 805964, DOI https://doi.org/10.1090/S0002-9947-1985-0805964-6
Wolfgang Dahmen and Charles A. Micchelli, Algebraic properties of discrete box splines, Constr. Approx. 3 (1987), no. 2, 209–221. MR 889556, DOI https://doi.org/10.1007/BF01890565
Wolfgang Dahmen and Charles A. Micchelli, The number of solutions to linear Diophantine equations and multivariate splines, Trans. Amer. Math. Soc. 308 (1988), no. 2, 509–532. MR 951619, DOI https://doi.org/10.1090/S0002-9947-1988-0951619-X
Wolfgang Dahmen and Charles A. Micchelli, On multivariate $E$-splines, Adv. Math. 76 (1989), no. 1, 33–93. MR 1004486, DOI https://doi.org/10.1016/0001-8708%2889%2990043-1
Thomas W. Hungerford, Algebra, Graduate Texts in Mathematics, vol. 73, Springer-Verlag, New York-Berlin, 1980. Reprint of the 1974 original. MR 600654
Rong Qing Jia, Linear independence of translates of a box spline, J. Approx. Theory 40 (1984), no. 2, 158–160. MR 732698, DOI https://doi.org/10.1016/0021-9045%2884%2990026-1
Rong Qing Jia, Local linear independence of the translates of a box spline, Constr. Approx. 1 (1985), no. 2, 175–182. MR 891538, DOI https://doi.org/10.1007/BF01890029

---, Symmetric magic squares and multivariate splines, preprint 1991.

Rong Qing Jia, Sherman Riemenschneider, and Zuowei Shen, Dimension of kernels of linear operators, Amer. J. Math. 114 (1992), no. 1, 157–184. MR 1147721, DOI https://doi.org/10.2307/2374741

---, Solvability of systems of linear operator equations, preprint 1991.

Amos Ron, Exponential box splines, Constr. Approx. 4 (1988), no. 4, 357–378. MR 956173, DOI https://doi.org/10.1007/BF02075467
Amos Ron, A necessary and sufficient condition for the linear independence of the integer translates of a compactly supported distribution, Constr. Approx. 5 (1989), no. 3, 297–308. MR 996932, DOI https://doi.org/10.1007/BF01889611
Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062
Charles C. Sims, Abstract algebra, John Wiley & Sons, Inc., New York, 1984. A computational approach. MR 746307

Th. Skolem, Diophantine Gleichungen, Chelsea, New York, 1950.

R. Stanley, Enumerative combinatorics, Vol. 1, Wadsworth, Belmont, Calif., 1986.