On the Approximation of Continuous Functions by Fourier–Legendre Sums Original Research Article

Let {Xn}∞0be the orthonormal system of Legendre polynomials on [−1, 1]. Forf∈C[−1, 1] letSn(f) (n+1∈N) be thenth partial sum of the Fourier–Legendre series of the functionf. Some refinements of the classical inequality[formula]involving best approximation inLp-norms are discussed. For a class of examples we obtain better order estimates than those that can be derived from (1). Furthermore, we show that the results are best possible in a certain sense. It turns out that only in two particular cases (p=43andp=4) there is no proof of optimality of the results. In conclusion, we give without proof a generalization of the main theorem to the ultraspherical case.