Orthogonal polynomials in \(L^ 1\)-approximation
Suppose that \(b_ 1,b_ 2,...,b_ n\) are given real numbers. The author considers the problem of determining a function h on \([0,\pi]\) with \(\ell\) changes of sign on \([0,\pi]\) (\(\ell \geq n)\), such that \(\int^{\pi}_{0}\sin (k\phi)h(\phi)d\phi =b_ k\) \((k=1,...,n)\). It is shown that the points at which the function h changes sign depend (under appropriate conditions for the numbers \(\{b_ k\}\) in a certain manner on some orthogonal polynomials. As an application the author considers Posse's problem: for which a and b \((1<a<b)\) is there a polynomial \(\tilde q_ n(x)=x^ n+..\). such that for every monic polynomial \(q_ n\) of degree n \(\int^{1}_{0} |\tilde q_ n| + (-1)^ n\int^{b}_{a}\tilde q_ n\leq \int^{1}_{a} | q_ n| + (- 1)^ n\int^{b}_{a}q_ n;\) furthermore find a minimizing polynomial when the conditions are satisfied. Posse solved this problem with the help of elliptic functions; the present author expresses the minimizing polynomials in terms of orthogonal polynomials on two intervals. Further applications are given in \(L_ 1\)-approximation on several intervals, in particular it is shown that there is a close connection between \(L_ 1\)- ,L\({}_ 2\)- and \(L_{\infty}\)-approximation with respect to a suitable weight on two intervals.
scientific article; zbMATH DE number 4046355