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
HTML articles powered by AMS MathViewer

by Rong Qing Jia

Trans. Amer. Math. Soc. 340 (1993), 179-198

DOI: https://doi.org/10.1090/S0002-9947-1993-1159194-6

PDF | Request permission

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.

References

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 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 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 10.1007/BF02786872

Discrete box splines and refinement algorithms

1

Wolfgang Dahmen, On multivariate $B$-splines, SIAM J. Numer. Anal. 17 (1980), no. 2, 179–191. MR 567267, DOI 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 10.1016/0024-3795(83)80015-9

Subdivision algorithms for the generation of box spline surfaces

1

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 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 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 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 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 10.1016/0001-8708(89)90043-1
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 10.1016/0021-9045(84)90026-1
Rong Qing Jia, Local linear independence of the translates of a box spline, Constr. Approx. 1 (1985), no. 2, 175–182. MR 891538, DOI 10.1007/BF01890029

Symmetric magic squares and multivariate splines

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 10.2307/2374741

Solvability of systems of linear operator equations

Amos Ron, Exponential box splines, Constr. Approx. 4 (1988), no. 4, 357–378. MR 956173, DOI 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 10.1007/BF01889611
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
Charles C. Sims, Abstract algebra, John Wiley & Sons, Inc., New York, 1984. A computational approach. MR 746307

Diophantine Gleichungen

Enumerative combinatorics