The paper investigates approximation error of two-layer feedforward Fourier Neural Networks (FNNs). Such networks are motivated by the properties series. Several implementations FNNs were proposed since 1980s: Gallant and White, Silvescu, Tan, Zuo Cai, Liu. main focus our work is Silvescu's FNN, because its activation function does not fit into category networks, where linearly transformed inpu...

We present a randomized approximation algorithm for counting contingency tables, m × n non-negative integer matrices with given row sums R = (r1, . . . , rm) and column sums C = (c1, . . . , cn). We define smooth margins (R,C) in terms of the typical table and prove that for such margins the algorithm has quasi-polynomial NO(lnN) complexity, where N = r1 + · · · + rm = c1 + · · · + cn. Various ...

In contrast to Fourier series, these finite power sums are over the angles equally dividing the upper-half plane. Moreover, these beautiful and somewhat surprising sums often arise in analysis. In this note, we extend the above results to the power sums as shown in identities (17), (19), (25), (26), (32), (33), (34), (35), and (36) and in the appendix. The method is based on the generating func...

In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem. Let h be a linear combination of exponentials with real frequencies. Determine all frequencies, all coefficients, and the number of summands if finitely many perturbed, uniformly sampled ...

We consider the problem of approximating functions by sums of wave packets. Our objective is to find sparse decompositions of image functions, over a finite range of scales. We also address the naturally connected task of approximating the wavefront set, computationally. We formulate the problem in terms of Hankel operators, Hankel matrices and their low-rank approximations, and develop an alge...

The paper presents upper estimates of the error of weighted and unweighted simultaneous approximation by the Bernstein operators and their iterated Boolean sums. The estimates are stated in terms of the Ditzian-Totik modulus of smoothness or appropriate K-functionals. AMS classification: 41A10, 41A17, 41A25, 41A28, 41A35, 41A36.