Pointwise error estimate of the Legendre expansion: The known and unknown features

Error estimation for the pointwise error of partial expansion of the Legendre series of a given function is a classical problem. Asymptotic error estimates exist expressed by the modulus of continuity of the given function, for functions of bounded variation, and for both analytic and piecewise analytic functions. These estimates are all asymptotic in the sense that the order of the partial expansion tends to infinity. It is interesting that in spite of many attempts the estimates are not optimal. This fact is demonstrated here, and the optimal estimate for the piecewise analytic functions is formulated as a hypothesis, which in turn is validated computationally. Further, in most applications the polynomial orders are relatively small compared to those in the asymptotic range. The existing theory does not address the behaviour of the preasymptotic error. Inherent difficulties addressing this question are discussed and illustrated here. (C) 2018 Elsevier B.V. All rights reserved.