On Approximation of Linear Functionals on L_p Spaces

[Ajit Dingankar]

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URL:
ftp://archive.cis.ohio-state.edu/pub/neuroprose/dingankar.linear-functionals.ps.Z

BiBTeX entry:
@ARTICLE{atd16,
	AUTHOR		= "Sandberg, I. W. and Dingankar, A. T.",
	TITLE		= "{On Approximation of Linear Functionals on
		 $L_p$ Spaces}",
	JOURNAL		= "IEEE Transactions on Circuits and
Systems-I: Fundamental Theory and Applications",
	VOLUME		= {},
	NUMBER		= {},
	PAGES		= {},
	YEAR		= "1995",
	}

        On Approximation of Linear Functionals on L_p Spaces
        ----------------------------------------------------

				ABSTRACT

In a recent paper certain approximations to continuous nonlinear
functionals defined on an $L_p$ space $ (1 < p < \infty) $ 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 $L_p$ may be approximated
uniformly by certain finite sums of integrals.

We also describe the relevance of this result to the approximation of
continuous nonlinear functionals with neural networks.