Pointwise Error Bounds for Legendre Approximations of Singular Functions

In this paper, we present pointwise error bounds for Legendre approximations of singular functions in fractional spaces. We begin by recalling the relevant fractional space definitions needed for the pointwise estimates. We then derive explicit and pointwise error bounds for Legendre approximations of functions with interior or endpoint singularities. New analytical techniques are developed to handle the case of endpoint singularities, and all coefficients in our estimates are given by explicit and computable formulas, allowing for a precise quantification of the singularity’s influence and clearly demonstrating the sharpness of the results. Numerical experiments are presented to support the theoretical results.