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 [Formula presented] 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.Ministry of Education (MOE)The research of the first author was supported by the National Natural Science Foundation of China (12271128). The research of the second author is partially supported by Singapore MOE AcRF Tier 1 Grant: RG15/21. The research of the third author was supported by the National Natural Science Foundation of China (11971131, 61873071, 51476047) and Natural Sciences Foundation of Heilongjiang Province (ZD2022A001)