Bernstein-type constants for approximation of |x|α by partial Fourier–Legendre and Fourier–Chebyshev sums

In this paper, we study the approximation of fα(x)=|x|α,α>0 in L∞[−1,1] by its Fourier–Legendre partial sum Sn(α)(x). We derive the upper and lower bounds of the approximation error in the L∞-norm that are valid uniformly for all n≥n0 for some n0≥1. Such an optimal L∞-estimate requires a judicious summation rule that can recover the lost half order if one uses a naive summation. Consequently, we can obtain the explicit Bernstein-type constant B∞(α)≔limn→∞nα‖fα−Sn(α)‖L∞=2Γ(α)π|sinαπ2|. Interestingly, using a similar argument, we can show that the Fourier–Chebyshev sum has the same Bernstein-type constant B∞(α) as the Legendre case.