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On Global Inverse Theorems of Szász and Baskakov Operators

Published online by Cambridge University Press:  20 November 2018

Z. Ditzian
Z. Ditzian*
Affiliation:
University of Alberta, Edmonton, Alberta
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The Szász and Baskakov approximation operators are given by

1.1

1.2

respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by

1.3

where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 2 , 01 April 1979 , pp. 255 - 263
Copyright
Copyright © Canadian Mathematical Society 1979

References

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