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

@article{Wang2020HowFD, title={How fast does the best polynomial approximation converge than Legendre projection?}, author={Haiyong Wang}, journal={ArXiv}, year={2020}, volume={abs/2001.01985} }

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. For differentiable functions such as piecewise analytic functions and functions of fractional smoothness…

## 5 Citations

Optimal error estimates for Legendre approximation of singular functions with limited regularity

- MathematicsArXiv
- 2020

This paper concerns optimal error estimates for Legendre polynomial expansions of singular functions whose regularities are naturally characterised by a certain fractional Sobolev-type space…

Spectral convergence of probability densities

- Mathematics, Computer ScienceArXiv
- 2021

This paper provides convergence rates for PDFs using colocation and Galerkin gPC methods in all dimensions, guaranteeing exponential rates for analytic maps and provides an alternative proof strategy based on optimal-transport techniques.

On the optimal rates of convergence of Gegenbauer projections

- Mathematics, Computer ScienceArXiv
- 2020

For piecewise analytic functions, it is demonstrated that the convergence rate of the Gegenbauer projection of degree $n$ is the same as that of the best approximation of the same degree when $\lambda\leq1$ and the former is slower than the latter by a factor of $n^{\lambda-1}$ when $\ lambda>1$.

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.

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:…

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