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 (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 we define a derivative in , which agrees with for smooth *f*, and prove several properties of .

**[1]**Hermann G. Burchard,*Splines (with optimal knots) are better*, Applicable Anal.**3**(1973/74), 309–319. MR**0399708**, https://doi.org/10.1080/00036817408839073**[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****[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****[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**, https://doi.org/10.1007/BF01817511**[5]**Carl de Boor,*On uniform approximation by splines*, J. Approximation Theory**1**(1968), 219–235. MR**0240519****[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****[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****[10]**Jean-Pierre Kahane,*Teoria constructiva de funciones*, Universidad de Buenos Aires, Buenos Aires, 1961 (Spanish). MR**0145254****[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****[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**, https://doi.org/10.1007/BF02788653**[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****[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****[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:
https://doi.org/10.1090/S0002-9947-1977-0481758-4

Article copyright:
© Copyright 1977
American Mathematical Society