Kronecker product bases and their applications in approximation theory

The Kronecker product is widely utilized to construct higher-dimensional spaces from lower-dimensional ones, making it an indispensable tool for efficiently analyzing multi-dimensional systems across various fields. This paper investigates the representation of analytic functions within hyper-elliptical regions through infinite series expansions involving sequences of Kronecker product bases of polynomials. Additionally, we examine the growth order and type and $T_{\rho }$-property of series composed of Kronecker product bases that represent entire functions. We also delve into the convergence properties of Kronecker product bases associated with special functions, including Bessel, Chebyshev, Bernoulli, Euler, and Gontcharoff polynomials. The obtained results extend and enhance the existing findings of such representations in hyper-spherical regions.