Some special series in ultraspherical polynomials and their approximation properties

Using the explicit form of a limiting ultraspherical series $\sum_{k=0}^\infty f_k^{-1}\widehat P_k^{-1}(x)$, which was established by us in [1], we consider new, more general, special series in ultraspherical Jacobi polynomials and their approximation properties. We show that as an approximation tool, these series compare favourably with Fourier series in Jacobi polynomials. At the same time, they admit a simple construction, which in important particular cases enables one to use the fast Fourier transform for the numerical realization of their partial sums.