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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

1. Becker, M., Global approximation theorem for Szâsz-Mirakjan and Baskakov operators in polynomial weight spaces, (To appear).Google Scholar
2. Becker, M., Kucharski, D. and Nessel, R., Global approximation theorems for the Szâsz-Mirakjan operators in exponential weight space, (To appear).Google Scholar
3. Berens, H. and Lorentz, G. G., Inverse theorems for Bernstein polynomials, Indiana Univ. Math. Jour.. 21 (1972), 693–708.Google Scholar
4. Butzer, P. L. and Scherer, K., Jackson and Bernstein-type inequalities for families of commutative operators in Banach spaces, Jour, of Approximation. 5 (1972), 308–343.Google Scholar
5. Ditzian, Z. and May, C. P., Lp saturation and inverse theorems for modified Bernstein polynomials, Indiana Math. Jour., (1976), 733–751.Google Scholar
6. Lupas, A., Some properties of the linear positive operators (J), Mathematica (Cluj), 9 (1967), 77–83.Google Scholar
7. May, C. P., Saturation and inverse theorems for combinations of a class of exponential type operators, Can. J. Math. 28 (1976), 1224–1250.Google Scholar
8. Martini, R., On the approximation of functions together with their derivatives by certain linear positive operators, Indag. Math. 31 (1969), 473–481.Google Scholar