# zbMATH — the first resource for mathematics

Homology of smooth splines: Generic triangulations and a conjecture of Strang. (English) Zbl 0718.41017
Summary: For $$\Delta$$ a triangulated d-dimensional region in $${\mathbb{R}}^ d$$, let $$S^ r_ m(\Delta)$$ denote the vector space of all $$C_ r$$ functions F on $$\Delta$$ that, restricted to any simplex in $$\Delta$$, are given by polynomials of degree at most m. We consider the problem of computing the dimension of such spaces. We develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds $$\Delta$$ in the plane, getting lower bounds on the dimension of $$S^ r_ m(\Delta)$$ for all r. For $$r=1$$, we prove a conjecture of Strang concerning the generic dimension of the space of $$C^ 1$$ splines over a triangulated manifold in $${\mathbb{R}}^ 2$$. Finally, we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region.

##### MSC:
 41A15 Spline approximation 65D07 Numerical computation using splines 05A15 Exact enumeration problems, generating functions 52A37 Other problems of combinatorial convexity 55N25 Homology with local coefficients, equivariant cohomology 57Q15 Triangulating manifolds
##### Keywords:
nonsimplicial decompositions of a plane region
Full Text:
##### References:
 [1] P. Alfeld, On the dimension of multivariate piecewise polynomials, Numerical analysis (Dundee, 1985) Pitman Res. Notes Math. Ser., vol. 140, Longman Sci. Tech., Harlow, 1986, pp. 1 – 23. · Zbl 0655.41010 [2] Peter Alfeld, A case study of multivariate piecewise polynomials, Geometric modeling, SIAM, Philadelphia, PA, 1987, pp. 149 – 159. [3] Peter Alfeld, Bruce Piper, and L. L. Schumaker, An explicit basis for \?\textonesuperior quartic bivariate splines, SIAM J. Numer. Anal. 24 (1987), no. 4, 891 – 911. · Zbl 0658.65008 · doi:10.1137/0724058 · doi.org [4] Peter Alfeld, Bruce Piper, and L. L. Schumaker, Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness \? and degree \?\ge 4\?+1, Comput. Aided Geom. Design 4 (1987), no. 1-2, 105 – 123. Special issue on topics in computer aided geometric design (Wolfenbüttel, 1986). · Zbl 0668.41011 · doi:10.1016/0167-8396(87)90028-8 · doi.org [5] P. Alfeld, B. Piper, and L. L. Schumaker, Spaces of bivariate splines on triangulations with holes, Proceedings of China-U.S. Joint Conference on Approximation Theory (Hangzhou, 1985), 1987, pp. 1 – 10. · Zbl 0687.41017 [6] Peter Alfeld and L. L. Schumaker, The dimension of bivariate spline spaces of smoothness \? for degree \?\ge 4\?+1, Constr. Approx. 3 (1987), no. 2, 189 – 197. · Zbl 0646.41008 · doi:10.1007/BF01890563 · doi.org [7] Richard H. Bartels, Splines in interactive computer graphics, Numerical analysis (Dundee, 1983) Lecture Notes in Math., vol. 1066, Springer, Berlin, 1984, pp. 1 – 29. · Zbl 0544.65009 · doi:10.1007/BFb0099515 · doi.org [8] Louis J. Billera, The algebra of continuous piecewise polynomials, Adv. Math. 76 (1989), no. 2, 170 – 183. · Zbl 0703.13015 · doi:10.1016/0001-8708(89)90047-9 · doi.org [9] Charles K. Chui and Ren Hong Wang, On smooth multivariate spline functions, Math. Comp. 41 (1983), no. 163, 131 – 142. · Zbl 0542.41008 [10] Charles K. Chui and Ren Hong Wang, Multivariate spline spaces, J. Math. Anal. Appl. 94 (1983), no. 1, 197 – 221. · Zbl 0526.41027 · doi:10.1016/0022-247X(83)90014-8 · doi.org [11] Ph. Ciarlet, Lectures on the finite element method, Tata Institute of Fundamental Research, Bombay, 1975. Notes by S. Kesavan, Akhil Ranjan and M. Vanninathan. [12] R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc. 49 (1943), 1 – 23. · Zbl 0063.00985 [13] Henry Crapo and Juliette Ryan, Réalisations spatiales des scènes linéaires, Structural Topology 13 (1986), 33 – 68. Dual French-English text. [14] 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. · Zbl 0559.41011 [15] R. Haas, Dimension and bases for certain classes of splines: a combinatorial and homological approach, Ph.D. thesis, Cornell Univ., August 1987. [16] James R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. · Zbl 0673.55001 [17] John Morgan and Ridgway Scott, A nodal basis for \?\textonesuperior piecewise polynomials of degree \?\ge 5, Math. Comput. 29 (1975), 736 – 740. · Zbl 0307.65074 [18] -, The dimension of the space of $${C^1}$$ piecewise polynomials, unpublished manuscript, 1975. [19] Larry L. Schumaker, On the dimension of spaces of piecewise polynomials in two variables, Multivariate approximation theory (Proc. Conf., Math. Res. Inst., Oberwolfach, 1979) Internat. Ser. Numer. Math., vol. 51, Birkhäuser, Basel-Boston, Mass., 1979, pp. 396 – 412. · Zbl 0461.41006 [20] Larry L. Schumaker, Bounds on the dimension of spaces of multivariate piecewise polynomials, Rocky Mountain J. Math. 14 (1984), no. 1, 251 – 264. Surfaces (Stanford, Calif., 1982). · Zbl 0601.41034 · doi:10.1216/RMJ-1984-14-1-251 · doi.org [21] Edwin H. Spanier, Algebraic topology, Springer-Verlag, New York-Berlin, 1981. Corrected reprint. · Zbl 0145.43303 [22] Peter F. Stiller, Certain reflexive sheaves on \?$$^{n}$$_\? and a problem in approximation theory, Trans. Amer. Math. Soc. 279 (1983), no. 1, 125 – 142. · Zbl 0523.14019 [23] Gilbert Strang, Piecewise polynomials and the finite element method, Bull. Amer. Math. Soc. 79 (1973), 1128 – 1137. · Zbl 0285.41009 [24] Gilbert Strang, The dimension of piecewise polynomial spaces, and one-sided approximation, Conference on the Numerical Solution of Differential Equations (Univ. Dundee, Dundee, 1973) Springer, Berlin, 1974, pp. 144 – 152. Lecture Notes in Math., Vol. 363. [25] Ren Hong Wang, Structure of multivariate splines, and interpolation, Acta Math. Sinica 18 (1975), no. 2, 91 – 106 (Chinese). · Zbl 0358.41004 [26] Neil L. White and Walter Whiteley, The algebraic geometry of stresses in frameworks, SIAM J. Algebraic Discrete Methods 4 (1983), no. 4, 481 – 511. · Zbl 0542.51022 · doi:10.1137/0604049 · doi.org [27] Walter Whiteley, A matroid on hypergraphs, with applications in scene analysis and geometry, Discrete Comput. Geom. 4 (1989), no. 1, 75 – 95. · Zbl 0656.05047 · doi:10.1007/BF02187716 · doi.org [28] -, A matrix for splines, J. Approx. Theory (to appear). [29] -, The analogy between multivariate splines and hinged panel structures, preprint, Champlain Regional College, June 1986. [30] Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. · Zbl 0081.26501
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.