ABSTRACT
The univariate Cardaliaguet-Euvrard (2.1) operators were first introduced and studied extensively in [62], where the authors, among many other interesting things, proved that these operators converge uniformly on compacta, to the unit over continuous and bounded functions. The univariate squashing operator (2.26) was motivated and inspired by the "squashing function?' and generated Theorem 6 of [62]. The work in [62] is qualitative where the used bell-shaped function is general. However, the work presented here, though greatly motivated by [62], is quantitative and the used bell-shaped and "squashing" functions are of compact support. We give a series of inequalities giving close upper bounds to the errors in approximating the unit operator by the above univariate neural network induced operators. All involved constants are well determined. These are mainly pointwise estimates involving the first modulus of continuity of the engaged continuous function or its derivatives. We also present some Lp-related estimates.
