One important topic in approximation theory is the approximation by linear operators of convolution type. In many cases of interest the kernels are, for example, integrable functions or bounded measures, thus belong to certain Banach algebras. As the approximation is governed by a detailed study of the kernels, it seems to be natural to look on this section of approximation theory from the point of view of abstract Banach algebras, particularly since the Gelfand transform may be used as a substitute for integral transforms, the most effective tool in the classical setting.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bragard, G.K. — Nessel, R.J., Teilbarkeitssätze in Banach-Algebren mit Anwendungen auf lineare Approximationsprozesse (to appear in Forschungsberichte des Landes Nordrhein-Westfalen).Google Scholar
  2. [2]
    Butzer, P.L. — Nessel, R.J., Fourier Analysis and Approximation I, Basel/New York, Birkhäuser/Academic Press 1971.CrossRefGoogle Scholar
  3. [3]
    Butzer, P.L. — Trebels, W., Operateurs de Gauß-Weierstraß et de Cauchy-Poissons et conditions lipschitz — Termes dans L 1(E n), C.R. Acad. Sci. Paris 268 (1969), 700–703.Google Scholar
  4. [4]
    Hewitt, E. — Ross, K.A., Abstract Harmonic Analysis I, II, Berlin, Springer 1963, 1972.Google Scholar
  5. [5]
    Loomis, L.H., An Introduction to Abstract Harmonie Analysis, New York, van Nostrand 1953.Google Scholar
  6. [6]
    Shapiro, H.S., A Tauberian theorem related: to approximation theory, Acta Math. 120 (1960), 279–292.CrossRefGoogle Scholar
  7. [7]
    Shapiro, H.S., Topics in Approximation Theory, Lecture Notes 187, Berlin, Springer 1971.Google Scholar
  8. [8]
    Šreider, Yu.A., The structure of maximal ideals in rings of measures with convolution, Mat. Sb. (N.S.) 27 (69) (1950), 297–318 (≡Amer. Math. Soc. Transi. 81 (1953), 365-391).Google Scholar

Copyright information

© Springer Basel AG 1974

Authors and Affiliations

  • G. K. Bragard
    • 1
  • R. J. Nessel
    • 1
  1. 1.Lehrstuhl A für MathematikRheinisch-Westfälische Technische HochschuleAachenDeutschland

Personalised recommendations