Polynomial approximation with doubling weights

Abstract

A nonnegative function ${w \in \mathbb{L}_1[-1, 1]}$ is called a doubling weight if there is a constant L such that ${w(2I) \leqq L w(I)}$, for all intervals ${I \subset [-1, 1]}$, where 2I denotes the interval having the same center as I and twice as large as I, and ${w(I) := \int_I w(u) du}$. In this paper, we establish direct and inverse results for weighted approximation by algebraic polynomials in the ${\mathbb{L}_p, 0 < p \leqq \infty}$, (quasi)norm weighted by ${w_n := \rho_n {(x)}^{-1} \int_{x - \rho_n {(x)}}^{x + \rho_n {(x)}} w(u) du}$, where ${\rho_n {(x)} := n^{-1} \sqrt{1 - x^2} + n^{-2}}$ and w is a doubling weight.

Among other things, we prove that, for a doubling weight ${w, 0 < p \leqq \infty, r \in \mathbb{N}_0}$, and ${0 < \alpha < r + 1 - 1/\lambda_p}$, we have

$$(\ast) \qquad E_n{(f)}_{p, w_n} = O(n^{-\alpha}) \iff \omega_\varphi^{r + 1}(f, n^{-1})_{p, w_n} = O(n^{-\alpha}),$$

where ${\lambda_p := p}$ if ${0 < p < \infty, \lambda_p :=1}$ if ${p = \infty, ||f||_{p, w} := \big(\int_{-1}^1 |f(u)|^p w(u) du \big)^{1/p}, ||f||_{\infty, w} := {\rm ess sup}_{u \in [-1, 1]} \left(|f(u)| w(u)\right), \omega_\varphi^r{(f, t)}_{p, w} := \sup_{0 < h \le t} \left\|\Delta_{h\varphi(\cdot)}^r (f, \cdot)\right\|_{p, w}, E_n{(f)}_{p, w} := \inf_{P_n \in \Pi_n} ||f - P_n||_{p, w}}$, and ${\Pi_n}$ is the set of all algebraic polynomials of degree ${\leqq n - 1}$.

We will also introduce classes of doubling weights ${\mathcal{W}^{\delta, \gamma}}$ with parameters ${{\delta, \gamma \geqq 0}}$ that are used to describe the behavior of ${w_n(x)/w_m(x)}$ for ${m \leqq n}$. It turns out that every class ${\mathcal{W}^{\delta, \gamma}}$ with ${(\delta, \gamma) \in \Upsilon := \{(\delta, \gamma) \in \mathbb{R}^2 \mid \delta \geqq 1, \gamma \geqq 0, \delta + \gamma \geqq 2\}}$ contains all doubling weights w, and for each pair ${(\delta, \gamma) \notin \Upsilon}$, there is a doubling weight not in ${\mathcal{W}^{\delta, \gamma}}$. We will establish inverse theorems and equivalence results similar to (${\ast}$) for doubling weights from classes ${\mathcal{W}^{\delta, \gamma}}$. Using the fact that ${1 \in \mathcal{W}^{0, 0}}$, we get the well known inverse results and equivalences of type (${\ast}$) for unweighted polynomial approximation as an immediate corollary.

Equivalence type results involving related K-functionals and realization type results (obtained as corollaries of our estimates) are also discussed.

Finally, we mention that (${\ast}$) closes a gap left in the paper by G. Mastroianni and V. Totik [13], where (${\ast}$) was established for ${p = \infty}$ and ${\omega_\varphi^{r + 2}}$ instead of ${\omega_\varphi^{r + 1}}$ (it was shown there that, in general, (${\ast}$) is not valid for ${p = \infty}$ if ${\omega_\varphi^{r + 1}}$ is replaced by ${\omega_\varphi^{r}}$).