Degree of rational functions of best approximation in \(L_ p(R^ m)\) (Q1326029)

Let \(A \subset \mathbb{R}^ m\) be a measurable set with \(\text{mes}_ m (A)>0\), and \(\rho(X)\) defined on \(A\) be a weight function. Denote by \(L_ p (A,\rho)\) \((1 \leq p < \infty)\) the set of measurable, complex- valued functions on \(A\) such that \(\| f \|_{L_ p (A,\rho)} : = (\int_ A \rho (X) | f(x) |^ pdx)^{{1 \over p}} < \infty\). Let \(R_{n,k} : = \{r(x) = s(x)/T(x) : s \in P_ n\), \(T \in P_ k\), \(n,k \in Z_ t\) fixed\}, \({\mathcal P}_ n\) consists of all polynomials in \(x_ 1, \dots, x_ m\) of degree \(\leq n\) with respect to the totality of variables. A typical result of this paper is the following. Theorem 2. If \(1<p< \infty\), \(f \in L_ p (A, \rho)\), \(\text{Re} f \overline \in R_{n,k}\), and a pair \((n,k)\) is admissible, i.e., \(R_{n,k} \cap L_ p (A,\rho)\) contains nonzero elements, then the best \(L_ p\)-approximation element of \(f\) is normal in \(R_{n+2,k+2}\) (i.e., it belongs to \(R_{n+2,k+2}\backslash R_{n+1,k+1)}\) or in \(R_{n+1,k+1}\); if \(A\) is bounded, then such a rational function is normal in \(R_{n+1,k+1}\).