An extremal problem for algebraic polynomials with zero mean value on an interval (Q1277526)

Let \(P_n\) be the set of all algebraic polynomials of degree \(n\) with real coefficients. For a given weight \(h\), where the function \(h\) is summable and nonnegative on \(I= [-1,1]\) and nonzero on a set of positive measure from \(I\), we define \[ P^0_n(h)= \Biggl\{p_n\in P_n: \int^1_{-1} p_n(x) h(x)ds= 0\Biggr\}. \] Let \(P^*_n(h)\) be a subset of \(P^0_n(h)\) of such polynomials \(p_n\) that the set \(\{x\in I: p_n(x)> 0\}\) is a single interval and \(P^1_n(h)\) be a subset of \(P^0_n(h)\) of such polynomials \(p_n\) which change sign on \((-1,1)\) exactly once. In the paper the following results are proved: Theorem 1. Let \(E(p_n)= \{x\in I: p_n(x)\geq 0\}\) and let \(\mu(p_n)= \text{mes }E(p_n)\). Thus \[ i_n(h)= \inf\{\mu(p_n): p_n\in P^0_n(h)\}= i^*_n(h)= \min\{\mu(p_n): p_n\in P^*_n(h)\}. \] Theorem 2. If a weight \(h\) has the \(\theta\)-property, that is \(h\) is positive, continuous in \(I\) and for any \(\theta\in (0,1)\) the functions \(h(t- 1)/h(\theta t-1)\), \(h(1- t)/h(1-\theta t)\), are nondecreasing with respect to \(t\) in \([0,2]\), then \[ i_n(h)= \inf\{\mu(p_n): p_n\in P^0_n(h)\}= j_n(h)= \min\{\mu(p_n): p_n\in P^1_n(h)\}. \]