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

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 judiciou...

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Bibliographic Details
Main Authors: Liu, Wenjie, Wang, Li-Lian, Wu, Boying
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2023
Subjects:
Science::Mathematics
Bernstein Constants
Legendre Polynomial Expansion
Online Access:https://hdl.handle.net/10356/169935
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Institution: Nanyang Technological University
Language: English
Description
Summary: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.

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