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The universal approximation property of feedforward neural networks (FNNs) is the basis for all FNNs applications. In almost all existing FNN approximation studies, it is always assumed that the activation function of the network satisfies certain conditions, such as sigmoid or bounded continuity. This paper focuses on building simple architecture FNNs with approximation ability by constructing an activation function. First, for any continuous function defined on [0, 1] and an arbitrary accuracy ϵ > 0, an FNN is constructed, which has only one neuron and a fixed weight 1 and can approximate the function with an accuracy of ϵ. Thereafter, we study the representation of FNNs for polynomial functions by constructing a proper activation function. It is proved that any algebraic polynomial with the degree n can be represented by an FNN. Further, a piecewise function is constructed as an activation function of an FNN such that the FNN represents the famous Bernstein polynomials. [ABSTRACT FROM AUTHOR]

This paper studies the construction and approximation of quasi-interpolation for spherical scattered data. First of all, a kind of quasi-interpolation operator with Gaussian kernel is constructed to approximate the spherical function, and two Jackson type theorems are established. Second, the classical Shepard operator is extended from Euclidean space to the unit sphere, and the error of approximation by the spherical Shepard operator is estimated. Finally, the compact supported kernel is used to construct quasi-interpolation operator for fitting spherical scattered data, where the spherical modulus of continuity and separation distance of scattered sampling points are employed as the measurements of approximation error. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper discusses local uniform error estimates for spherical basis functions (SBFs) interpolation, where error bounds for target functions are restricted on spherical cap. The discussion is first carried out in the native space associated with the smooth SBFs, which is generated by a strictly positive definite zonal kernel. Then, the smooth SBFs are embedded in a larger space that is generated by a less smooth kernel, and for the target functions outside the original native space, the local uniform error estimates are established. Finally, some numerical experiments are given to illustrate the theoretical results. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]