On Approximation of Linear Functionals on L p Spaces

In a recent paper certain approximations to continuous nonlinear functionals defined on an Lp space (1 ! p ! 1) are shown to exist. These approximations may be realized by sigmoidal neural networks employing a linear input layer that implements finite sums of integrals of a certain type. In another recent paper similar approximation results are obtained using elements of a general class of continuous linear functionals. In this note we describe a connection between these results by showing that every continuous linear functional on a compact subset of Lp may be approximated uniformly by certain finite sums of integrals. I. INTRODUCTION One of the earliest results in the area of neural networks is the proposition that any continuous real function defined on a compact subset of IR k (k an arbitrary positive integer) can be approximated arbitrarily well using a single-hiddenlayer network with sigmoidal nonlinearities (see, for example, [2]). Among other results in the literature conce..