Spectral synthesis deals with the description of translation invariant function spaces. It turns out that the basic building blocks of this description are the exponential monomials, which are built up from exponential functions and polynomial functions. The author collaborated with Laczkovich [Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 1, 103–120] proved that spectral synthesis holds on an Abelian group if and only if the torsion free rank of the group is finite. The author [Aequationes Math. 70 (2005), no. 1-2, 122–130] showed that the torsion free rank of an Abelian group is strongly related to the properties of polynomial functions on the group. Here we show that spectral synthesis holds on an Abelian group if and only if the ring of polynomial functions on the group is Noetherian.

Our current investigation is primarily motivated by the application of special polynomials in Geometric Function Theory (GFT). This paper aims to utilize (M, N)-Lucas polynomials to estimate the initial coefficient bounds for a subclass of bi-univalent functions consisting of normalized analytic functions. We then derive the famous Fekete-Szegö inequality estimate. We also establish connections between our results and those examined in previous investigations.

A large class of nonlinear dynamical systems can be presented by polynomial nonlinear systems. Necessary conditions for optimal control of nonlinear systems are reduced to solve nonlinear differential equations with boundary conditions which their solutions in general can not be analytically obtained. Numerical solutions and spectral methods have been used in order to solve these systems. In this paper a numerical method is presented by using spectral hybrid functions for solving polynominal nonlinear, systems A numerical example is included to demonstrate the validity and applicability of the method and a comparison is made by existing results.        

In this paper we obtain the explicit form of the best uniform polynomial approximations out of Pn of two classes of rational functions using properties of Chebyshev polynomials. In this way we present some new theorems and lemmas. Some examples will be given to support the theoretical results.

Unknown data relations can describe a lot of complex systems through a partial differential equation solution of a multi-parametric function approximation. Common artificial neural network techniques of a pattern classification or function approximation in general are based on whole-pattern similarity relations of trained and tested data samples. It applies input variables of only absolute interval values, which may cause problems by far various training and testing data ranges. Differential polynomial neural network is a new type of neural network developed by the author, which constructs and resolves an unknown general partial differential equation, describing a system model of dependent variables. It creates a sum of fractional polynomial terms, defining partial mutual derivative changes of input variables combinations. This type of regression is based on learned generalized data relations. It might improve dynamic system models a standard time-series prediction, as the character of relative data allows to apply a wider range of input interval values than defined by the trained data. Also the characteristics of differential equation solutions facilitate a great variety of model forms.

In this paper, we consider a new class of analytic functions in the unit disk using polynomials of order  . We give some sufficient conditions for functions belonging to this class.

This investigation deals with two new subclasses of analytic and bi-univalent functions defined by Bernoulli polynomial. In this paper, coefficient estimation and Fekete-Szegö problems are solved for these newly defined function subclasses. In addition, certain remarks are indicated for the subclasses of bi-starlike and bi-convex functions.

The nonlinear conjugate gradient method solves issues of the frame: minimizef(x), x ∈,ℜ,employing an iterative plot, x(k+1) = x(k) + α, kd(k), where f is a non-polynomial function. We utilized two variants of the optimum line search namely, direct and indirect methods, to compute the step-length in this paper. Both line searches yielded great outcome when employed to a few unconstrained non-polynomial test functions.