Approximation of analytic functions of exponential derived and integral bases in Fréchet spaces

Abstract

The theory of basic sets (bases) of polynomials has a significant role in mathematics and its applications, e.g., approximation theory, mathematical physics, Geometry, and partial differential equations. The interest of the present work focused on the expansion of analytic functions into BPs. Given a sequence of a base of polynomialsThe expansion of an analytic function as a basic series ∑ began with the papers by Whittaker and Cannon [14,15,30,31] about 90 years ago. Basic series generalize Taylor series, where can be Legendre, Laguerre, Chebyshev, Hermite, Bessel, Bernoulli and Euler polynomials (see [1,4,5,10,11,21]). This theory found a lot of applications, mainly in the theory of functions depending on one or several complex variables as well as in the approximation of solutions of differential equations or matrix functions.The topic of derivative BPs in one complex variable has been studied early (see [26,27,28]), the searchers considered the disks in the complex plane C. For several complex variables (see [16,17,19,24,25]), the representation domains are polycyclinderical, hyperspherical and hyperelliptical regions. Recently,in [12,35] the

Representation in Fréchet Spaces of Hyperbolic Theta and Integral Operator Bases for Polynomials

“…An exhaustive study of BPs in Clifford and complex analysis are to be found in [6, 9 , 13, 25, 27, 29, 38, 42, 44, 47]. Also, the study of BPs in complex conformable fractional derivative is present in the papers [26,30,48,49].…”