Approximate interpolation by a class of neural networks in Lebesgue metric

In this paper, a class of approximate interpolation neural networks is constructed to approximate Lebesgue integrable functions. It is showed that the networks can arbitrarily approximate any p-th Lebesgue integrable function in Lebesgue metric as long as the number of hidden nodes is sufficiently large. The relation among the approximation speed, the number of hidden nodes, the interpolation sample and the smoothness of the target function is also revealed by designing the Steklov mean function and the modulus of smoothness of f. The obtained results are helpful in studying the problem of approximation complexity of interpolation neural networks in Lebesgue metric.

1 Introduction

Throughout this paper, and denote the natural numbers set and the set of real numbers, respectively, and, for any positive integer denotes the dimensional Euclidean space. Let denote a set of distinct vectors, and a set of real numbers. Then

$$\{(x_{0}, f_{0}), (x_{1}, f_{1}), \ldots, (x_{n}, f_{n})\}\eqno{\hbox{(1)}}$$ is called a set of interpolation sample, and is called a node system of interpolation.