Best local approximations in \(L^ p(\mu)\) (Q921324)

Let \(\mu\) be a positive Borel measure on the unit ball B in \(R^ n\) with \(\mu (B)=1\). Define \(\mu_{\epsilon}\) at \(E\subset B\) by \(\mu_{\epsilon}(E)=\mu (\epsilon E)/\mu (\epsilon B)\). Denote by \(P_{m,\mu}f\) the unique element of \(\pi_ m\) (the class of real polynomials of degree at most m) satisfying \(\int_{B}| f- P_{m,\mu}f|^ pd\mu =\inf_{P\in \pi_ m}\int_{B}| f- P|^ pd\mu.\) Given that \(\lim_{\epsilon \to 0+}\int_{B}x^ k_ jd\mu_{\epsilon}\) exists for \(j=1,...,n\) and \(k=0,1,2,..\). (here \(x=(x_ 1,...,x_ j,...,x_ n)\in R^ n)\) we study the limiting behaviour as \(\epsilon \to 0+\) of \[ (E_{\epsilon}f)(t):=\epsilon^{- m-1}[f(\epsilon t)-(P_{m,\mu_{\epsilon}}f(\epsilon \cdot))(\epsilon t)],\quad | t| \leq 1, \] when f belongs to the subspace \(t^ p_{m,\mu}\) of \(L^ p(\mu)\) first considered by Calderon and Zygmund. Our results generalize those of \textit{R. A. Macías} and \textit{F. Zó} [J. Approximation Theory 42, 181-192 (1984; Zbl 0567.41011)].