Erdös-Turán-type theorems on piecewise smooth curves and arcs
Let \(\sigma\) be a signed measure on a Jordan curve \(L\), then \(\sup|\sigma (J)|\) -- here the supremum is taken over all measurable subarcs \(J\) of \(L\) -- is called the discrepancy of \(\sigma\) on \(L\); further let \(\mu_L\) be the equilibrium measure on \(L\). Let \(p\) be a monic polynomial of degree \(n\) and let \(\nu_p\) be the corresponding normalized zero counting measure. For a piecewise smooth \(L\) the authors prove upper estimates of \(\sigma=\mu_L- \nu_p\) in terms of the logarithmic potential associated to \(\sigma\). As an application they obtain well-known results on Fekete points and extremal points of best uniform approximants.
scientific article; zbMATH DE number 980683
scientific article