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 𝑇𝜌 - property and the mode of increase of the
polynomial bases as defined by hyperbolic Theta operator bases and hyperbolic integral operator bases. 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.
2000 Mathematics Subject Classification: 30 E 10, 30 D 10, 41 A 10, 26 A 33
Keywords Hyperbolic Theta operator, Basic series, Bases of polynomials, Effectiveness,
Order and Type, Fréchet space.