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On the degree of convergence of piecewise polynomial approximation on optimal meshes


Author: H. G. Burchard
Journal: Trans. Amer. Math. Soc. 234 (1977), 531-559
MSC: Primary 41A15
DOI: https://doi.org/10.1090/S0002-9947-1977-0481758-4
MathSciNet review: 0481758
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Abstract: The degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain F-spaces $ {\mathbf{N}}_0^{p,n}$ (introduced for this purpose in [2]). We obtain two o-results and use pairs of inequalities of Bernstein- and Jackson-type to prove several direct and converse theorems. For f in $ {\mathbf{N}}_0^{p,n}$ we define a derivative $ {D^{n,\sigma }}f$ in $ {L^\sigma },\sigma = {(n + {p^{ - 1}})^{ - 1}}$, which agrees with $ {D^n}f$ for smooth f, and prove several properties of $ {D^{n,\sigma }}$.


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  • [1] H. G. Burchard, Splines (with optimal knots) are better, J. Appl. Anal. 3 (1974), 309-319. MR 0399708 (53:3551)
  • [2] H. G. Burchard and D. F. Hale, Piecewise polynomial approximation on optimal meshes, J. Approximation Theory 14 (1975), 128-147. MR 51 # 10957. MR 0374761 (51:10957)
  • [3] P. L. Butzer and R. J. Nessel, Fourier analysis and approximation, Vol. 1, Academic Press, New York, 1971. MR 0510857 (58:23312)
  • [4] P. L. Butzer and K. Scherer, On the fundamental approximation theorems of D. Jackson, S. N. Bernstein and theorems of M. Zamansky and S. B. Stečkin, Aequationes Math. 3 (1969), 170-185. MR 41 #8897. MR 0264301 (41:8897)
  • [5] C. de Boor, On uniform approximation by splines, J. Approximation Theory 1 (1968), 219-235. MR 39 #1866. MR 0240519 (39:1866)
  • [6] -, Good approximation by splines with variable knots, Spline Functions and Approximation Theory (A. Meir and A. Sharma, editors), Birkhäûser, Basel, 1972, pp. 57-72. MR 0403169 (53:6982)
  • [7] D. S. Dodson, Optimal order approximation by polynomial spline functions, Thesis, Purdue Univ., 1972.
  • [8] N. Dunford and J. Schwartz, Linear operators. Vol. I, Interscience, New York, 1958. MR 22 #8302.
  • [9] G. Freud and V. A. Popov, Certain questions connected with approximation by spline-functions and polynomials, Studia Sci. Math. Hungar. 5 (1970), 161-171. (Russian) MR 42 #2225. MR 0267323 (42:2225)
  • [10] J. P. Kahane, Teoria constructiva de funciones, Cursos y Sem. Mat., Fasc. 5, Univ. Buenos Aires, Buenos Aires, 1961. MR 26 #2728. MR 0145254 (26:2787)
  • [11] D. E. McClure, Nonlinear segmented function approximation and analysis of line patterns, Tech. Report, Div. Appl. Math., Brown Univ. 1973.
  • [12] G. M. Phillips, Error estimates for best polynomial approximations, Approximation Theory (Proc. Sympos., Lancaster, 1969), A. Talbot, Editor, Academic Press, London, 1970, pp. 1-6. MR 43 #3703. MR 0277970 (43:3703)
  • [13] H. B. Curry and I. J. Schoenberg, On Pólya frequency functions. IV: The fundamental spline functions and their limits, J. Analyse Math. 17 (1966), 71-107. MR 36 # 1884. MR 0218800 (36:1884)
  • [14] C. R. de Boor, Good approximation with variable knots. II, Conf. on the Numerical Solution of Differential Equations (Dundee, 1973), Lecture Notes in Math., vol. 363, Springer-Verlag, Berlin and New York, 1974, pp. 12-20. MR 0431606 (55:4603)
  • [15] H. G. Burchard and D. F. Hale, Direct and converse theorems for piecewise polynomial approximation on optimal partitions, Notices Amer. Math. Soc. 20 (1973), A-277. Abstract #73T-B100.
  • [16] H. G. Burchard, Degree of convergence of piecewise polynomial approximation on optimal meshes. II, Notices Amer. Math. Soc. 21 (1974), A-639. Abstract #719-B11.
  • [17] Ju. A. Brudnyĭ, Spline approximation, and functions of bounded variation, Dokl. Akad. Nauk SSSR 215 (1974), 511-515 = Soviet Math. Dokl. 15 (1974), 518-521. MR 52 #6249. MR 0385386 (52:6249)
  • [18] J. Bergh and J. Peetre, On the spaces $ {V_p}\;(0 < p \leqslant \infty )$, Tech. Report 1974:7, Dept. of Math., Univ. of Lund, 1974.

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DOI: https://doi.org/10.1090/S0002-9947-1977-0481758-4
Article copyright: © Copyright 1977 American Mathematical Society

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