On the degree of convergence of piecewise polynomial approximation on optimal meshes
Author:
H. G. Burchard
Journal:
Trans. Amer. Math. Soc. 234 (1977), 531559
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 Fspaces (introduced for this purpose in [2]). We obtain two oresults and use pairs of inequalities of Bernstein and Jacksontype to prove several direct and converse theorems. For f in we define a derivative in , which agrees with for smooth f, and prove several properties of .
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H. G. Burchard and D. F. Hale, Direct and converse theorems for piecewise polynomial approximation on optimal partitions, Notices Amer. Math. Soc. 20 (1973), A277. Abstract #73TB100.
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H. G. Burchard, Degree of convergence of piecewise polynomial approximation on optimal meshes. II, Notices Amer. Math. Soc. 21 (1974), A639. Abstract #719B11.
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A. Brudnyĭ, Spline approximation, and functions of bounded
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J. Bergh and J. Peetre, On the spaces , Tech. Report 1974:7, Dept. of Math., Univ. of Lund, 1974.
 [1]
 H. G. Burchard, Splines (with optimal knots) are better, J. Appl. Anal. 3 (1974), 309319. MR 0399708 (53:3551)
 [2]
 H. G. Burchard and D. F. Hale, Piecewise polynomial approximation on optimal meshes, J. Approximation Theory 14 (1975), 128147. 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), 170185. MR 41 #8897. MR 0264301 (41:8897)
 [5]
 C. de Boor, On uniform approximation by splines, J. Approximation Theory 1 (1968), 219235. 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. 5772. 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 splinefunctions and polynomials, Studia Sci. Math. Hungar. 5 (1970), 161171. (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. 16. 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), 71107. 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, SpringerVerlag, Berlin and New York, 1974, pp. 1220. 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), A277. Abstract #73TB100.
 [16]
 H. G. Burchard, Degree of convergence of piecewise polynomial approximation on optimal meshes. II, Notices Amer. Math. Soc. 21 (1974), A639. Abstract #719B11.
 [17]
 Ju. A. Brudnyĭ, Spline approximation, and functions of bounded variation, Dokl. Akad. Nauk SSSR 215 (1974), 511515 = Soviet Math. Dokl. 15 (1974), 518521. MR 52 #6249. MR 0385386 (52:6249)
 [18]
 J. Bergh and J. Peetre, On the spaces , Tech. Report 1974:7, Dept. of Math., Univ. of Lund, 1974.
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DOI:
http://dx.doi.org/10.1090/S00029947197704817584
PII:
S 00029947(1977)04817584
Article copyright:
© Copyright 1977
American Mathematical Society
