The FC-Gram trigonometric polynomial approximation of a non-periodic function that interpolates the function on equispaced grids was introduced by Bruno and Lyon in 2010 [13]. Since then, the approximation algorithm and its further refinements have been used extensively in numerical methods for solving various PDE-based problems, and it has had impressive success in handling challenging configurations. While much computational evidence exists in the literature confirming the rapid convergence of FC-Gram approximations, a theoretical convergence analysis has remained open. In this paper, we study a modified FC-Gram algorithm where the implicit least-squares-based periodic extensions of the Gram polynomials are replaced with an explicit extension utilizing two-point Hermite polynomials. This modification brings in two significant advantages - (i) as the extensions are known explicitly, the need to use computationally expensive precomputed extension data is eliminated, which, in turn, facilitates seamlessly changing the extension length, and (ii) allows for establishing provable error bounds for the modified approximations. Through a variety of computational experiments, we show that the numerical convergence rates are consistent with those predicted by the theory. Moreover, to demonstrate the efficacy of this approximation strategy in scientific computation applications, we utilize it for obtaining high-order accurate numerical solutions of problems involving differential equations.
