In this study, we present a novel definition for hyperbolic Theta operator bases (HTOBs) and hyperbolic integral operator bases (HIOBs) within the context of complex calculus. We employ the constructed HTOBs and HIOBs on a specific base of polynomials (BPs) across diverse convergence regions within Fréchet spaces. Consequently, we explore the correlation between the approximation properties of the resulting base and the original one. Furthermore, we derive insights into the -property and the mode of increase of the polynomial bases as defined by HTOBs and HIOBs. The investigation extends to various bases of special polynomials, including Chebyshev, Bessel, Gontcharaff, Euler and Bernoulli, polynomials, ensuring the robustness and applicability of the obtained results.