On smooth multivariate spline functions

Authors:
Charles K. Chui and Ren Hong Wang

Journal:
Math. Comp. **41** (1983), 131-142

MSC:
Primary 41A15; Secondary 41A63

DOI:
https://doi.org/10.1090/S0025-5718-1983-0701629-1

MathSciNet review:
701629

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper the dimensions of bivariate spline spaces with simple cross-cut grid partitions are determined and expressions of their basis functions are given. Consequently, the closures of these spaces over all partitions of the same type can be determined. A somewhat more detailed study on bivariate splines with rectangular grid partitions is included. The results in this paper can be applied to problems on interpolation and approximation by bivariate spline functions.

**[1]**R. E. Barnhill, G. Birkhoff & W. J. Gordon, "Smooth interpolation in triangles,"*J. Approx. Theory*, v. 8, 1973, pp. 114-128. MR**0368382 (51:4623)****[2]**R. K. Beatson, "Convex approximation by splines,"*SIAM J. Math. Anal.*(To appear.) MR**617714 (82h:41012)****[3]**G. Birkhoff & H. Garabedian, "Smooth surface interpolation,"*J. Math. Phys.*, v. 39, 1960, pp. 258-268. MR**0119387 (22:10151)****[4]**C. de Boor, "On calculating with*B*-splines,"*J. Approx. Theory*, v. 6, 1972, pp. 50-62. MR**0338617 (49:3381)****[5]**C. de Boor, "Splines as linear combination of*B*-splines," in*Approximation Theory*II (G. G. Lorentz, C. K. Chui, L. L. Schumaker, Eds.), Academic Press, New York, 1976, pp. 1-47.**[6]**C. de Boor,*A Practical Guide to Splines*. Springer-Verlag, New York, 1978. MR**507062 (80a:65027)****[7]**C. de Boor & R. DeVore, "Approximation by smooth multivariate splines." (In manuscript.)**[8]**C. K. Chui, P. W. Smith & J. D. Ward, "Degree of approximation by monotone splines,"*SIAM J. Math. Anal.*, v. 11, 1980, 436-447. MR**572194 (81h:41019)****[9]**C. K. Chui, P. W. Smith & J. D. Ward, "Monotone approximation by spline functions," in*Quantitative Approximation*(R. A. DeVore and K. Scherer, Eds.), Academic Press, New York, 1980. MR**588172 (81k:41006)****[10]**C. K. Chui & R. H. Wang, "Bases of bivariate spline spaces with cross-cut grid partitions,"*J. Math. Res. Exposition*, v. 2, 1982, pp. 1-3. MR**658038 (83e:41013)****[11]**S. A. Coons, "Surface patches and*B*-spline curves," in*Computer Aided Geometric Design*(R. E. Barnhill and R. F. Riesenfeld, Eds.), Academic Press, New York, 1974.**[12]**W. Dahmen, "On multivariate*B*-splines,"*SIAM J. Numer. Anal.*, v. 17, 1980, pp. 179-190. MR**567267 (81c:41020)****[13]**W. Dahmen, R. DeVore & K. Scherer, "Multidimensional spline approximations,",*SIAM J. Numer. Anal.*, v. 17, 1980, pp. 380-402. MR**581486 (81j:41015)****[14]**W. Dahmen & C. A. Micchelli,*On Limits of Multivariate B-Splines*, MRC Technical Summary Report #2114, University of Wisconsin, 1980. MR**632464 (82j:41013)****[15]**R. A. DeVore, "Monotone approximation by splines,"*SIAM J. Math. Anal.*, v. 8, 1977, pp. 891-905. MR**0510725 (58:23259)****[16]**T. Dupont & R. Scott, "Polynomial approximation of functions in Sobolev spaces,"*Math. Comp.*, v. 34, 1980, pp. 441-463. MR**559195 (81h:65014)****[17]**W. J. Gordon, "Blending-function methods of bivariate and multivariate interpolation and approximation,"*SIAM J. Numer. Anal.*, v. 8, 1971, pp. 158-177. MR**0282498 (43:8209)****[18]**C. A. Hall, "Bicubic interpolation over triangles,"*J. Math. Mech.*, v. 19, 1969, pp. 1-11. MR**0245211 (39:6523)****[19]**C. A. Micchelli, "A constructive approach to Kergin interpolation in : Multivariate*B*-splines and Lagrange interpolation,"*Rocky Mountain J. Math.*(To appear.) MR**590212 (84i:41002)****[20]**C. A. Micchelli, "On numerically efficient methods for computing multivariate*B*-splines," in*Multivariate Approximation Theory*(W. Schempp and K. Zeller, Eds.), ISNM 51, Birkhauser-Verlag, Basel, 1979. MR**560673 (81g:65017)****[21]**M. Munteanu & L. L. Schumaker, "Direct and inverse theorems for multi-dimensional spline approximation,"*Indiana Univ. Math. J.*, v. 23, 1973, pp. 461-470. MR**0338643 (49:3407)****[22]**I. J. Schoenberg, "Contributions to the problem of approximation of equidistant data by analytic functions,"*Quart. Appl. Math.*, v. 4, 1946, pp. 45-99, 112-141.**[23]**M. H. Schultz,*Spline Analysis*, Prentice-Hall, Englewood Cliffs, N.J., 1973. MR**0362832 (50:15270)****[24]**L. L. Schumaker, "Fitting surfaces to scattered data," in*Approximation Theory*II (G. G. Lorentz, C. K. Chui and L. L. Schumaker, Eds.), Academic Press, New York, 1976. MR**0426369 (54:14312)****[25]**L. L. Schumaker, "On the dimension of spaces of piecewise polynomials in two variables," in*Multivariate Approximation Theory*(W. Schempp and K. Zeller, Eds.), ISBN 51, Birkhauser-Verlag, Basel, 1979. MR**560683 (81d:41011)****[26]**G. Strang, "The dimension of piecewise polynomials and one-sided approximation," in*Proc. Conf. Numerical Solution of Differential Equations*(Dundee 1973), Lecture Notes in Math., #365, Springer-Verlag, Berlin and New York, 1974, pp. 144-152. MR**0430621 (55:3626)****[27]**G. Strang, "Piecewise polynomials and finite element methods,"*Bull. Amer. Math. Soc.*, v. 79, 1973, pp. 1128-1137. MR**0327060 (48:5402)****[28]**R. J. Walker,*Algebraic Curves*, Princeton Univ. Press, Princeton, N.J., 1950. MR**0033083 (11:387e)****[29]**R. H. Wang, "The structural characterization and interpolation for multivariate splines,"*Acta Math. Sinica*, v. 18, 1975, pp. 91-106. MR**0454458 (56:12709)****[30]**R. H. Wang, "On the analysis of multivariate splines in the case of arbitrary partition,"*Sci. Sinica*(*Math.*I), 1979, pp. 215-226. MR**662201 (84e:41013)****[31]**R. H. Wang, "On the analysis of multivariate splines in the case of arbitrary partition. II,"*Num. Math. J. Chinese Univ.*, v. 2, 1980, pp. 78-81. MR**594890 (83h:41011)****[32]**R. H. Wang, S. Z. Liang & Y. S. Chou,*Approximation of Functions of Several Variables*(in Chinese), Science Press, Peking. (To appear.)

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0701629-1

Keywords:
Multivariate spline functions,
total degree,
*B*-splines,
conformality condition,
basis,
cross-cuts

Article copyright:
© Copyright 1983
American Mathematical Society