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Some Recent Applications of Functional Analysis to Approximation Theory

In: Zum Werk Leonhard Eulers

Author

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  • P. L. Butzer

    (Rheinisch-Westfälische Technische Hochschule Aachen, Lehrstuhl A für Mathematik)

Abstract

A major portion of approximation theory is concerned with the approximation of functions by polynomials or by sequences {Tn} of linear operators, more specifically with the connections between the structural properties of the function f being approximated and the convergence per se and/or rate of convergence of ‖Tn (f) — f ‖ to zero for n → ∞. in particular, the wide area of approximation theory and its applications is devoted to the convergence per se and the rate of vonvergence of, for example, (a) the best trigonometric approximation of a given function, (b) the partial sums of the Fourier series of a function to the function. itself, (c) the solution of Dirichlet’s problem for the unit disk to the given boundary value, (d) the Whittaker — Shannon sampling series expansion of a duration-limited function to the functioninquestion, (e) the sums occuring in the weak law of large numbers in probability theory.

Suggested Citation

  • P. L. Butzer, 1984. "Some Recent Applications of Functional Analysis to Approximation Theory," Springer Books, in: Eberhard Knobloch & Ilppo Simo Louhivaara & Jörg Winkler (ed.), Zum Werk Leonhard Eulers, pages 133-155, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-7121-1_7
    DOI: 10.1007/978-3-0348-7121-1_7
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