On smooth multivariate spline functions
Authors:
Charles K. Chui and Ren Hong Wang
Journal:
Math. Comp. 41 (1983), 131142
MSC:
Primary 41A15; Secondary 41A63
MathSciNet review:
701629
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Abstract: In this paper the dimensions of bivariate spline spaces with simple crosscut 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.
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 [1]
 R. E. Barnhill, G. Birkhoff & W. J. Gordon, "Smooth interpolation in triangles," J. Approx. Theory, v. 8, 1973, pp. 114128. 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. 258268. MR 0119387 (22:10151)
 [4]
 C. de Boor, "On calculating with Bsplines," J. Approx. Theory, v. 6, 1972, pp. 5062. MR 0338617 (49:3381)
 [5]
 C. de Boor, "Splines as linear combination of Bsplines," in Approximation Theory II (G. G. Lorentz, C. K. Chui, L. L. Schumaker, Eds.), Academic Press, New York, 1976, pp. 147.
 [6]
 C. de Boor, A Practical Guide to Splines. SpringerVerlag, 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, 436447. 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 crosscut grid partitions," J. Math. Res. Exposition, v. 2, 1982, pp. 13. MR 658038 (83e:41013)
 [11]
 S. A. Coons, "Surface patches and Bspline curves," in Computer Aided Geometric Design (R. E. Barnhill and R. F. Riesenfeld, Eds.), Academic Press, New York, 1974.
 [12]
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 [13]
 W. Dahmen, R. DeVore & K. Scherer, "Multidimensional spline approximations,", SIAM J. Numer. Anal., v. 17, 1980, pp. 380402. MR 581486 (81j:41015)
 [14]
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 [15]
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 [16]
 T. Dupont & R. Scott, "Polynomial approximation of functions in Sobolev spaces," Math. Comp., v. 34, 1980, pp. 441463. MR 559195 (81h:65014)
 [17]
 W. J. Gordon, "Blendingfunction methods of bivariate and multivariate interpolation and approximation," SIAM J. Numer. Anal., v. 8, 1971, pp. 158177. MR 0282498 (43:8209)
 [18]
 C. A. Hall, "Bicubic interpolation over triangles," J. Math. Mech., v. 19, 1969, pp. 111. MR 0245211 (39:6523)
 [19]
 C. A. Micchelli, "A constructive approach to Kergin interpolation in : Multivariate Bsplines and Lagrange interpolation," Rocky Mountain J. Math. (To appear.) MR 590212 (84i:41002)
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 C. A. Micchelli, "On numerically efficient methods for computing multivariate Bsplines," in Multivariate Approximation Theory (W. Schempp and K. Zeller, Eds.), ISNM 51, BirkhauserVerlag, Basel, 1979. MR 560673 (81g:65017)
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 M. Munteanu & L. L. Schumaker, "Direct and inverse theorems for multidimensional spline approximation," Indiana Univ. Math. J., v. 23, 1973, pp. 461470. MR 0338643 (49:3407)
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 I. J. Schoenberg, "Contributions to the problem of approximation of equidistant data by analytic functions," Quart. Appl. Math., v. 4, 1946, pp. 4599, 112141.
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 M. H. Schultz, Spline Analysis, PrenticeHall, 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, BirkhauserVerlag, Basel, 1979. MR 560683 (81d:41011)
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 G. Strang, "The dimension of piecewise polynomials and onesided approximation," in Proc. Conf. Numerical Solution of Differential Equations (Dundee 1973), Lecture Notes in Math., #365, SpringerVerlag, Berlin and New York, 1974, pp. 144152. MR 0430621 (55:3626)
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 G. Strang, "Piecewise polynomials and finite element methods," Bull. Amer. Math. Soc., v. 79, 1973, pp. 11281137. MR 0327060 (48:5402)
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 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:
http://dx.doi.org/10.1090/S00255718198307016291
PII:
S 00255718(1983)07016291
Keywords:
Multivariate spline functions,
total degree,
Bsplines,
conformality condition,
basis,
crosscuts
Article copyright:
© Copyright 1983
American Mathematical Society
