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

Abstract

The approximation of an analytic function as a basic series of the following type where is a BPs, has been developed by the British Mathematician J. M. Whittaker in the early 1930 s [39]. J. M. Whittaker and B. Cannon [15,16,40,41] have obtained many results about approximation of analytic and entire functions by basic series (1.1). Numerous specific instances of polynomial series have undergone thorough examination. Taylor's series stands out as the simplest case, with other notable examples including expansions from interpolation theory and series involving polynomials such as Hermite, Legendre, Legendre, Euler, Bernoulli, Chebyshev, and Gontcharoff. Several scholars, including Makar [34], Mikhail [35], and Newns [37], have delved into the convergence properties of derivative and integral bases for a given set of BPs in a single complex variable within a disk centered at the origin. For multiple complex variables, as explored in [12,20,32,33], representation domains extend to polycylindrical, hyperspherical, and hyperelliptical regions.