Approximative properties of the Taylor-Abel-Poisson linear summation method Fourier series are considered for functions several variables, periodic with respect to hexagonal domain, in integral metric. In particular, direct and inverse theorems proved terms approximations by means K-functionals generated radial derivatives. Bernstein type inequalities L1-norm derivatives arbitrary order Poisson...

Modern convolutional networks, incorporating rectifiers and max-pooling, are neither smooth nor convex; standard guarantees therefore do not apply. Nevertheless, methods from convex optimization such as gradient descent and Adam are widely used as building blocks for deep learning algorithms. This paper provides the first convergence guarantee applicable to modern convnets, which furthermore ma...

Taylor Approximation for Chance Constrained Optimization Problems Governed by Partial Differential Equations with High-Dimensional Random Parameters

In this paper, we present a fourth order method for computing simple roots of nonlinear equations by using suitable Taylor and weight function approximation. The method is based on Weerakoon-Fernando method [S. Weerakoon, G.I. Fernando, A variant of Newton's method with third-order convergence, Appl. Math. Lett. 17 (2000) 87-93]. The method is optimal, as it needs three evaluations per iterate,...

Given a nonlinear system we determine a relation between controllability distributions defined for a nonlinear system and a Taylor series approximation of it. Special attention is given to this relation at the e,quilibrium. It is known from nonlinear control theory that the solvability conditions as well as the solutions to some control synthesis problems can be stated in terms of geometric con...

A CMOS current divider using a two-variable second-order Taylor series approximation is proposed. The approximation divider is realized with a compact circuit. The simulation results indicate that the approximation divider is feasible for the characteristics of sufficient accuracy, small distortion, and high bandwidth.

In this work, with combined belief propagation (BP), mean field (MF) and expectation propagation (EP), an iterative receiver is designed for joint phase noise (PN) estimation, equalization and decoding in a coded communication system. The presence of the PN results in a nonlinear observation model. Conventionally, the nonlinear model is directly linearized by using the first-order Taylor approx...

We propose a generic procedure to raise the approximation order of multivariate approximation schemes using supplementary derivative data. The procedure applies to all schemes that reproduce polynomials to a certain degree, including most common types of (quasi-) interpolation and moving least-squares. For an approximation scheme of orderm and a dataset that provides n supplementary orders of d...