Advertisement

Abstract

The present paper is, in spirit, a sequel to [5], but may be read independently of that paper. The underlying motive is to provide a satisfactory, and suitably quantitative, theory of “degree of approximation” for functions of several variables. In this section we discuss some ramifications of this general problem, and outline a program of investigation, of which some first steps are carried out in the following sections. Although, or perhaps because, our results are quite fragmentary, we hope they might stimulate interest in the present circle of questions.

Keywords

Linear Functional Absolute Constant Tauberian Theorem Underlying Motive Mild Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Boman, Supremum norm estimates for partial derivatives of functions of several variables.To appear.Google Scholar
  2. [2]
    Y. A. Brudni, Constructive characteristics of functions given on certain perfect sets on the real axis in “Investigations on Contemporary Problems of the Constructive Theory of Functions”, pp. 122–126. Fizmatgiz, Moscow 1961 (Russian).Google Scholar
  3. [3]
    D. J. Newman, Efficiency of polynomials on sequences. J. Approx. Theory 1 (1968), no. 1, 66–76.CrossRefGoogle Scholar
  4. [4]
    D. J. Newman and H. S. Shapiro, Some theorems on Cebysev approximation. Duke Math. J. 30 (1963), 673–682.CrossRefGoogle Scholar
  5. [5]
    D. J. Newman and H. S. Shapiro, Jackson’s theorem in higher dimensions, in “On Approximation Theory”. ISNM vol. 5. Birkhäuser, Basel 1964, 208–219.Google Scholar
  6. [6]
    S. M. Nikolski, On imbedding, continuation and approximation theorems for differentiable functions of several variables. Uspehi Mat. Nauk 16 (1961). no. 5 (101), 63–114 (Russian).Google Scholar
  7. [7]
    S. M. Nikolski, Inequalities for entire functions of finite type and applications to the theory of differentiable functions of several variables. Trudy Mat. Inst. Steklova 38 (1951), 244–278 (Russian).Google Scholar
  8. [8]
    J. Peetre, Reflections about Besov spaces. Mimeographed lecture notes, Lund (Swedish).Google Scholar
  9. [9]
    J. Peetre, On the theory of L p,λ spaces. To appear.Google Scholar
  10. [10]
    D. L. Ragozin, Approximation theory on compact manifolds and Lie groups, with applications to harmonic analysis. Dissertation, Harvard 1967.Google Scholar
  11. [11]
    H. S. Shapiro, Some Tauberian theorems with applications to approximation theory. Bull. Amer.Math. Soc. 74 (1968), 500–504.CrossRefGoogle Scholar
  12. [12]
    H. S. Shapiro, A Tauberian theorem related to approximation theory. Acta Math. 120, 3–4 (1968), 279–292.CrossRefGoogle Scholar
  13. [13]
    H. S. Shapiro, Smoothing and Approximation of Functions. Matscience Report 55, Madras 1967, mimeographed (a revised edition will appear in Van Nostrand’s Mathematical Studies series).Google Scholar
  14. [14]
    V. Tihomirov, Diameters of sets in function spaces and the theory of best approximation. Uspehi Mat. Nauk. 15 (1960), no. 3 (93), 81–120 (Russian).Google Scholar
  15. [15]
    A. Zygmund, Trigonometric Series, Vol. I. Cambridge Univ. Press 1959.Google Scholar

Copyright information

© Springer Basel AG 1969

Authors and Affiliations

  • Harold S. Shapiro
    • 1
  1. 1.Dept. of Math.University of MichiganUSA

Personalised recommendations