The Convolution Transform. By Hirschmann & Widder . Pp. x + 268. 45s. 1955. (Princeton University Press. London : Cumberlege)

In this paper a new approximation operator is introduced and its properties are studied. Special cases of this operator are the well-known Szasz power-series approximation operator and its generalization by D. Leviatan. The behaviour of the new approximation operator at points of continuity and discontinuity is investigated by using probabilistic tools as the Chebishev inequality and Liapounov’s central limit theorem. Such probabilistic methods of proof simplify the proofs and give better understanding of the approximation mechanism.

B. Levikson

Papers

A new approximation operator of the Arato-Renyi type