Constructive Approximation by Superposition of Sigmoidal Functions

In this paper, a constructive theory is developed for approximating functions of one or more variables by superposition of sigmoidal functions. This is done in the uniform norm as well as in the $L^p$ norm. Results for the simultaneous approximation, with the same order of accuracy, of a function and its derivatives (whenever these exist), are obtained. The relation with neural networks and radial basis functions approximations is discussed. Numerical examples are given for the purpose of illustration.     

单变量Sigmoidal型神经网络的逼近

引入了一种新的sigmoidal型神经网络,给出了其对连续函数逼近的点态和整体估计.结果表明这种新的神经网络算子具有多项式逼近所不能达到的很好的逼近速度.为了改进对光滑函数的逼近速度,我们进一步引入了一种新的神经网络的线性组合,并给出了这种组合逼近的点态估计和整体估计.最后给出了一个数值例子.     

Approximation of bandlimited functions by finite exponential sums

For a compact set K in ℝ ⁿ , let B ² _K be the set of all functions f ∈ L ²(ℝ²) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B ² _K by finite exponential sums

Degree of approximation by superpositions of a sigmoidal function

In this paper we study the degree of approximation by superpositions of a sigmoidal function. We mainly consider the univariate case. If f is a continuous function, we prove that for any bounded sigmoidal function σ, . For the Heaviside function H(x), we prove that . If f is a continuous function of bounded variation, we prove that and . For he Heaviside function, the coefficient 1 and the approximation orders are the best possible. We compare these results with the classical Jackson and Bernstein theorems, and make some conjectures for further study.     

On superpositions of continuous functions

We show that if Φ is an arbitrary countable set of continuous functions of n variables, then there exists a continuous, and even infinitely smooth, function ψ(x₁,...,x_n) such that ψ(x ₁, ...,x _n ) ?g [? (f ₁(x ₁, ... ,f _f (x _n ))] for any function ? from Φ and arbitrary continuous functions g and f_i, depending on a single variable.     

Approximation of the unit step function by a linear combination of exponential functions

We approximate the unit step function, which equals 1 if t ε [0, T] and equals 0 if t > T, by functions of the form ∑_{n = 1}^N Ax_n^N e^−λ_n^t/T, where each λ_n is a given positive constant. We find the coefficients A_n^(N) by minimizing the integrated square of the difference between the unit step function and the approximating function. We first solve the specialized case where each λ_n = n. The resulting sum can be shown to converge in the mean to the unit step function as N → ∞. The general case is then solved and some interesting properties of the numbers A_n^(N) are noted.     

Approximation of the function ¦x¦ by rational functions

We consider the problem of the approximation of the function ¦x¦ by rational functions. We make more precise the best approximation estimate obtained by A. P. Bulanov. We prove that for arbitrary positive integral n where A is an absolute constant.Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 163–171, July, 1974.The author thanks E. P. Dolzhenko for his help in preparing this paper.     

Approximation of analytic functions by Laguerre functions

We will solve the inhomogeneous Laguerre differential equation and apply this result to prove that if a function can be represented by a power series whose radius of convergence is larger than 1, then the function can be approximated, on the interval [0, 1), by a Laguerre function with an error bound C x for some constant C > 0.     

Approximation by harmonic functions

For a compact set we construct a restoring covering for the space of real-valued functions on which can be uniformly approximated by harmonic functions. Functions from restricted to an element of this covering possess some analytic properties. In particular, every nonnegative function , equal to 0 on an open non-void set, is equal to 0 on . Moreover, when , the algebra of complex-valued functions on which can be uniformly approximated by holomorphic functions is analytic. These theorems allow us to prove that if a compact set has a nontrivial Jensen measure, then contains a nontrivial compact set with analytic algebra .

     

Approximation of analytic functions by Legendre functions

We will solve the inhomogeneous Legendre’s differential equation and apply this result to estimate the error bound occurring when an analytic function is approximated by an appropriate Legendre function.     

Approximation of analytic functions by Hermite functions

We solve the inhomogeneous Hermite equation and apply this result to estimate the error bound occurring when any analytic function is approximated by an appropriate Hermite function.     

Approximation of analytic functions by Legendre functions

We will solve the inhomogeneous Legendre’s differential equation and apply this result to estimate the error bound occurring when an analytic function is approximated by an appropriate Legendre function.