H. N. Mhaskar

Approximation by superposition of sigmoidal and radial basis functions

Let @s: R -> R be such that for some polynomial P, @sP is bounded. We consider the linear span of the functions {@s(@l . (x - t)): @l, t @e R^s}. We prove that unless @s is itself a polynomial, it is possible to uniformly approximate any continuous function on R^s arbitrarily well on every compact subset of R^s by functions in this span. Under more specific conditions on @s, we give algorithms to achieve this approximation and obtain Jackson-type theorems to estimate the degree of approximation.

Where Does the Sup Norm of a Weighted Polynomial Live? (A Generalization of Incomplete Polynomials)

A characterization is given of the sets supporting the uniform norms of weighted polynomials [w(x)]nPn(x), wherePn is any polynomial of degree at mostn. The (closed) support ∑ ofw(x) may be bounded or unbounded; of special interest is the case whenw(x) has a nonempty zero setZ. The treatment of weighted polynomials consists of associating each admissible weight with a certain functional defined on subsets of ∑ —Z. One main result of this paper states that there is a unique compact set (independent ofn andPn) maximizing this functional that contains the points where the norms of weighted polynomials are attained. The distribution of the zeros of Chebyshev polynomials corresponding to the weights [w(x)]n is also studied. The main theorems give a unified method of investigating many particular examples. Applications to weighted approximation on the real line with respect to a fixed weight are included.

Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature

Geodetic and meteorological data, collected via satellites for example, are genuinely scattered and not confined to any special set of points. Even so, known quadrature formulas used in numerically computing integrals involving such data have had restrictions either on the sites (points) used or, more significantly, on the number of sites required. Here, for the unit sphere embedded in R q , we obtain quadrature formulas that are exact for spherical harmonics of a fixed order, have nonnegative weights, and are based on function values at scattered sites. To be exact, these formulas require only a number of sites comparable to the dimension of the space. As a part of the proof, we derive L 1 -Marcinkiewicz-Zygmund inequalities for such sites.

Neural networks for optimal approximation of smooth and analytic functions

We prove that neural networks with a single hidden layer are capable of providing an optimal order of approximation for functions assumed to possess a given number of derivatives, if the activation function evaluated by each principal element satisfies certain technical conditions. Under these conditions, it is also possible to construct networks that provide a geometric order of approximation for analytic target functions. The permissible activation functions include the squashing function (1 ex)1 as well as a variety of radial basis functions. Our proofs are constructive. The weights and thresholds of our networks are chosen independently of the target function; we give explicit formulas for the coefficients as simple, continuous, linear functionals of the target function.

Introduction to the theory of weighted polynomial approximation

Polynomial inequalities degree of approximation applications of potential theory Freud-type orthogonal polynomials.

On trigonometric wavelets

Wavelets in terms of sine and cosine functions are constructed for decomposing 2π-periodic square-integrable functions into different octaves and for yielding local information within each octave. Results on a simple mapping into the approximate sample space, order of approximation of this mapping, and pyramid algorithms for decomposition and reconstruction are also discussed.

Dimension-independent bounds on the degree of approximation by neural networks

Let φ be a univariate 2π-periodic function. Suppose that s ≥ 1 and f is a 2π-periodic function of s real variables. We study sufficient conditions in order that a neural network having a single hidden layer consisting of n neurons, each with an activation function φ, can be constructed so as to give a mean square approximation to f within a given accuracy ∈ n , independent of the number of variables. We also discuss the case in which the activation function φ is not 2π-periodic.

Approximation properties of a multilayered feedforward artificial neural network

We prove that an artificial neural network with multiple hidden layers and akth-order sigmoidal response function can be used to approximate any continuous function on any compact subset of a Euclidean space so as to achieve the Jackson rate of approximation. Moreover, if the function to be approximated has an analytic extension, then a nearly geometric rate of approximation can be achieved. We also discuss the problem of approximation of a compact subset of a Euclidean space with such networks with a classical sigmoidal response function.

Approximation properties of zonal function networks using scattered data on the sphere

AbstractA zonal function (ZF) network is a function of the form x↦∑k=1nck