# How much faster does the best polynomial approximation converge than Legendre projection?

@article{Wang2021HowMF, title={How much faster does the best polynomial approximation converge than Legendre projection?}, author={Haiyong Wang}, journal={Numerische Mathematik}, year={2021}, volume={147}, pages={481-503} }

We compare the convergence behavior of best polynomial approximations and Legendre and Chebyshev projections and derive optimal rates of convergence of Legendre projections for analytic and differentiable functions in the maximum norm. For analytic functions, we show that the best polynomial approximation of degree $n$ is better than the Legendre projection of the same degree by a factor of $n^{1/2}$. For differentiable functions such as piecewise analytic functions and functions of fractional…

## 7 Citations

Spectral convergence of probability densities for forward problems in uncertainty quantification

- Mathematics, Computer ScienceNumerische Mathematik
- 2022

. The estimation of probability density functions (PDF) using approximate maps is a fundamental building block in computational probability. We consider forward problems in uncertainty quantiﬁcation:…

Are best approximations really better than Chebyshev?

- Mathematics, Computer ScienceArXiv
- 2021

It is shown that when the underlying function has an algebraic singularity, the Chebyshev projection of degree n converges one power of n faster than its best counterpart at each point away from the singularity and both converge at the same rate at the singularities.

J un 2 02 1 Are best approximations really better than Chebyshev ?

- Mathematics, Computer Science
- 2021

It is shown that when the underlying function has an algebraic singularity, the Chebyshev projection of degree n converges one power of n faster than its best counterpart at each point away from the singularity and both converge at the same rate at the singularities.

New error bounds for Legendre approximations of differentiable functions

- Mathematics, Computer ScienceArXiv
- 2021

A sequence of Legendre-Gauss-Lobatto polynomials is introduced and their theoretical properties are proved to derive a new and explicit bound for the Legendre coefficients of differentiable functions and establish some explicit and optimal error bounds for Legendre projections in the L and L∞ norms.

Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation

- MathematicsAdv. Comput. Math.
- 2021

A new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation is presented, which can enrich the existing theory for p and hp methods for singular problems, and answer some open questions posed in some recent literature.

The mysteries of the best approximation and Chebyshev expansion for the function with logarithmic regularities

- Computer Science, MathematicsArXiv
- 2021

It is found that for the functions with logarithmic regularities, the pointwise errors of Chebyshev approximation are smaller than the ones of the best approximations except only in the very narrow boundaries at the same degree.

Optimal rates of convergence and error localization of Gegenbauer projections

- Mathematics
- 2020

It is shown that the rate of convergence of Gegenbauer projections is the same as that of best approximations under conditions of the underlying function is either analytic on and within an ellipse and λ ≤ 0 or differentiable andλ ≤ 1, where λ is the parameter in Geganbauer projection.

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