On Taylor-like Estimates for \(L^{2}\) Polynomial Approximations

Polynomial series approximations are a central theme in approximation theory. Two types of series, which are featured most prominently in pure and applied mathematics, are Taylor series expansions and expansions derived based on families of \(L^{2}-\)orthogonal polynomials on bounded intervals. The identity theorem implies that all such series agree on \(\mathbb{C}\) in the limit. This motivates an effort to derive properties, which relate the original function to a truncated series based on \(L^{2}-\)orthogonal polynomial approximation which hold on unbounded intervals. In particular, the Chebyshev series expansion of \(e^{x}\) is studied and an algebraic criterion which can be used to confirm bounds analogous to the Taylor series upper and lower bound estimates for \(x