Summary
An analog of the well-known Jackson-Bernstein-Zygmund theory on best approximation by trigonometric polynomials is developed for approximation methods which use piecewise polynomial functions. Interpolation and best approximation by polynomial splines, Hermite and finite element functions are examples of such methods. A direct theorem is proven for methods which are stable, quasi-linear and optimally accurate for sufficiently smooth functions. These assumptions are known to be satisfied in many cases of practical interest. Under a certain additional assumption, on the family of meshes, an inverse theorem is proven which shows that the direct theorem is sharp.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlberg, J. H., Nilson, E. N., Walsh, J. L.: The theory of splines and their applications. New York: Academic Press 1967
Aubin, J.-P.: Approximation of elliptic boundary value problems. New York: Wiley 1972
Azis, A. K.: The mathematical foundations of the finite element method with applications to partial differential equations. Proc. Symp. Univ. of Maryland, 1972. New York: Academic Press 1972
Brenner, P., Thomée, V., Wahlbin, L. B.: Besov spaces and applications to difference methods for initial value problems. Berlin-Heidelberg-New York: Springer 1975
Burchard, H. G., Hale, D. F.: Piecewise polynomial approximation on optimal meshes. J. Approximation Theory14, 128–147 (1975)
Butler, G., Richards, F.: AnLp-saturation theorem for splines. Canad. J. Math.24, 957–966 (1972)
Butzer, P. L., Scherer, K.: On the fundamental approximation theorems of D. Jackson, S. N. Bernstein and the theorems of M. Zamansky and S. B. Stečkin. Aequations Math.3, 170–185 (1969)
Butzer, P. L., Scherer, K.: On the fundamental theorems of approximation theory and their dual versions. J. Approximation Theory,3, 87–100 (1970)
Campanato, S.: Proprietà di hölderianità di alcune classi di funzioni. Ann. Scuola Norm. Sup. Pisa17, 175–188 (1963)
Campanato, S.: Proprietà di una famiglia di spazi funzionali. Ann. Scuola Norm. Sup. Pisa18, 93–140 (1964)
de Boor, C.: On uniform approximation by splines. J. Approximation Theory1, 219–235 (1968)
Gagliardo, E.: Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi in funzioni inn variabili. Rend. Sem. Mat., Univ. Padova27, 284–305 (1957)
Gaier, D.: Saturation bei Spline-Approximation und Quadraturen. Numer. Math.16, 129–140 (1970)
Grevholm, B.: On the structure of the spaces ℒ p, λ k .Math. Scand.26, 241–254 (1970)
Hedstrom, G. W., Varga, R. S.: Application of Besov spaces to spline approximation. J. Approximation Theory4, 295–327 (1971)
Hörmander, L.: Estimates for translation invariant operators in Lp-spaces. Acta Math.104, 93–140 (1960)
Hörmander, L.: Linear partial differential operators. Berlin-Göttingen-Heidelberg: Springer 1963
Löfström, J.: Besov spaces in the theory of approximations. Ann. Math. Pure Appl.85, 93–184 (1970)
McClure, D. E.: Nonlinear segmented function approximation and analysis of line patterns. Quart. Appl. Math.33, 1–37 (1975)
Munteanu, M. J., Schumaker, L. L.: Direct and inverse theorems for multidimensional spline approximation. Indiana Univ. Math. J.23, 461–470 (1973)
Nikolskii, S. M.: Approximation of functions of several variables and imbedding theorems. Berlin-Heidelberg-New York: Springer 1975
Nitsche, J.: Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens. Numer. Math.11, 346–348 (1968)
Nitsche, J.: Sätze von Jackson-Bernstein-typ für die Approximation mit Spline-Funktionen. Math. Z.109, 97–106 (1969)
Nitsche, J.: Umkehrsätze für Spline-Approximation. Compositio Math.21, 400–416 (1969)
Peetre, J.: Funderingar om Becov-rum. Lecture notes (Swedish), Dept. of Math., Lund (1966)
Peetre, J.: Sur les espaces de Becov. C. R. Acad. Sc. (Paris)264, 281–283 (1967)
Peetre, J.: Analysis in quasi-Banach space and approximation theory. Technical report, 1972:3, Dept. of Math., Lund (1972)
Peetre, J.: New thoughts on Besov spaces. Lecture notes, Dept. of Math., Duke Univ. (1974)
Rice, J. R.: On the degree of convergence of nonlinear spline approximation, pp. 349–365, In: I. J. Schoenberg, Approximation with special emphasis on spline functions. Proc. Symp. Univ. Wisconsin, 1969. New York: Academic Press 1969
Richards, F.: On the saturation class for spline functions. Proc. Amer. Math. Soc.33, 471–476 (1972)
Scherer, K.: Über die beste Approximation von Lp-Funktion durch Splines. Proc. Conf. Constructive Function Theory, Varna, pp.277–286 (1970)
Scherer, K.: On the best approximation of continuous functions by splines. SIAM J. Numer. Anal.7, 418–423 (1972)
Scherer, K.: Characterization of generalized Lipschitz classes by best approximation with splines. SIAM J. Numer. Anal.11, 283–304 (1974)
Schoenberg, I. J.: Approximation with special emphasis on spline functions. Proc. Symp. Univ. Wisconsin, 1969. New York: Academic Press 1969
Schultz, M. H.: Spline analysis. Englewood Cliffs, N. J.: Prentice Hall 1973
Shapiro, H. S.: Smoothing and approximation of functions. New York: van Nostrand Reinhold 1969
Stein, E. M.: The characterization of functions arising as potentials II. Bull. Amer. Math. Soc.68, 577–582 (1962)
Stein, E. M.: Singular integrals and differentiability properties of functions. Princeton: Princeton Univ. Press 1970
Strang, G.: The finite element method and approximation theory, pp. 547–583, In: B. Hubbard, Numerical solution of partial differential equations—II, Synspade 1970. London-New York: Academic Press 1971
Strang, G.: Approximation in the finite element method. Numer. Math.19, 81–98 (1972)
Strang, G., Fix, G. J.: An analysis of the finite element method. Englewood Cliffs, N. J.: Prentice-Hall 1973
Taibleson, M. H.: On the theory of Lipschitz spaces of distributions on euclideann-space. I. J. Math. Mech.,13, 407–480 (1964); Taibleson, M. H.: On the theory of Lipschitz spaces of distributions on euclideann-space. II. (ibid) J. Math. Mech.,14, 821–840 (1965). Taibleson, M. H.: On the theory of Lipschitz spaces of distributions on euclideann-space. III (ibid)15, 973–981 (1966)
Timan, A. F.: Theory of approximation of functions of a real variable. New York: McMillan 1973
Widlund, O.: Some results on best possible error bounds for finite element methods and approximation with piecewise polynomial functions. Lecture notes in Math., Vol. 228, pp. 253–263. Berlin-Heidelberg-New York: Springer 1971
Widlund, O.: Schauder and Lp estimates for linear elliptic systems, revisited. (to appear)
Zygmund, A.: Trigonometric series, Vol. I. Cambridge: Cambridge University Press 1959
Author information
Authors and Affiliations
Additional information
The work presented in this paper was supported by the ERDA Mathematics and Computing Laboratory, Courant Institute of Mathematical Sciences, New York University, under Contract E(11-1)-3077 with the Energy Research and Development Administration.