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Best polynomial approximation on the unit ball
Classical Analysis and ODEs · Submitted 18 Sep 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our main result shows that, for $s \in \mathbb{N}$, $$ E_n(f)_\mu \le c n^{-2s}[E_{n-2s}(\Delta^s f)_{\mu+2s} + E_{n}(\Delta_0^s f)_{\mu}], $$ where $\Delta$ and $\Delta_0$ are the Laplace and Laplace-Beltrami operators, respectively.
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Let $E_n(f)_\mu$ be the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\varpi_\mu, \mathbb{B}^d)$, where $\mathbb{B}^d$ is the unit ball in $\mathbb{R}^d$ and $\varpi_\mu(x) = (1-\|x\|^2)^\mu$ for $\mu > -1$. Our main result shows that, for $s \in \mathbb{N}$, $$ E_n(f)_\mu \le c n^{-2s}[E_{n-2s}(\Delta^s f)_{\mu+2s} + E_{n}(\Delta_0^s f)_{\mu}], $$ where $\Delta$ and $\Delta_0$ are the Laplace and Laplace-Beltrami operators, respectively. We also derive a bound when the right hand side contains odd order derivatives.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $E_n(f)_\mu$ be the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\varpi_\mu, \mathbb{B}^d)$, where $\mathbb{B}^d$ is the unit ball in $\mathbb{R}^d$ and $\varpi_\mu(x) = (1-\|x\|^2)^\mu$ for $\mu > -1$. | background 1.00 | 0.13 | 0.03 | asserted 0.74 |
| s1 | Our main result shows that, for $s \in \mathbb{N}$, $$ E_n(f)_\mu \le c n^{-2s}[E_{n-2s}(\Delta^s f)_{\mu+2s} + E_{n}(\Delta_0^s f)_{\mu}], $$ where $\Delta$ and $\Delta_0$ are the Laplace and Laplace-Beltrami operators, respectively. | result 1.00 | 0.96 | 0.98 | asserted 1.00 |
| s2 | We also derive a bound when the right hand side contains odd order derivatives. | result 0.76 | 0.95 | 0.03 | asserted 1.00 |
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Original abstract
Let $E_n(f)_\mu$ be the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\varpi_\mu, \mathbb{B}^d)$, where $\mathbb{B}^d$ is the unit ball in $\mathbb{R}^d$ and $\varpi_\mu(x) = (1-\|x\|^2)^\mu$ for $\mu > -1$. Our main result shows that, for $s \in \mathbb{N}$, $$ E_n(f)_\mu \le c n^{-2s}[E_{n-2s}(\Delta^s f)_{\mu+2s} + E_{n}(\Delta_0^s f)_{\mu}], $$ where $\Delta$ and $\Delta_0$ are the Laplace and Laplace-Beltrami operators, respectively. We also derive a bound when the right hand side contains odd order derivatives.
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