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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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] Hermann G. Burchard, Splines (with optimal knots) are better, Applicable Anal. 3 (1973/74), 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), no. 2, 128–147. MR 0374761 (51 #10957)
  • [3] Paul L. Butzer and Rolf J. Nessel, Fourier analysis and approximation, Academic Press, New York-London, 1971. Volume 1: One-dimensional theory; Pure and Applied Mathematics, Vol. 40. 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 0264301 (41 #8897)
  • [5] Carl de Boor, On uniform approximation by splines, J. Approximation Theory 1 (1968), 219–235. MR 0240519 (39 #1866)
  • [6] Carl de Boor, Good approximation by splines with variable knots, Spline functions and approximation theory (Proc. Sympos., Univ. Alberta, Edmonton, Alta., 1972) Birkhäuser, Basel, 1973, pp. 57–72. Internat. Ser. Numer. Math., Vol. 21. 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. Fraĭd and V. A. Popov, Certain questions connected with approximation by spline-functions and polynomials, Studia Sci. Math. Hungar. 5 (1970), 161–171 (Russian). MR 0267323 (42 #2225)
  • [10] Jean-Pierre Kahane, Teoria constructiva de funciones, Universidad de Buenos Aires, Buenos Aires] 1961 (1961), 111 (Spanish). 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) Academic Press, London, 1970, pp. 1–6. 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 0218800 (36 #1884)
  • [14] Carl de Boor, Good approximation by splines with variable knots. II, Conference on the Numerical Solution of Differential Equations (Univ. Dundee, Dundee, 1973) Springer, Berlin, 1974, pp. 12–20. Lecture Notes in Math., Vol. 363. 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), 511–513 (Russian). 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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0481758-4
PII: S 0002-9947(1977)0481758-4
Article copyright: © Copyright 1977 American Mathematical Society