Coefficients of polynomials of restricted growth on the real line (Q1266128)

The author shows that under mild restrictions on \(\phi\), a continuous even function on \(\mathbb R\) with positive values, there exist decreasing sequences \(x_1,\dots,x_m\) and \(y_1,\dots,y_{m-1}\) of symmetrically located points on \(\mathbb R\) and corresponding polynomials \(P\) and \(Q\) of degrees \(m-1\) and \(m\), respectively, satisfying \(| P(x)| \leq\phi(x)^m, \;| Q(x)| \leq\phi(x)^m,\;x\in \mathbb R\), such that for any polynomial \(p\) of degree at most \(m\), (a) if \(| p(x_j)| \leq\phi(x_j)^m\) for \(j=1,\dots,m\), then \(| p^{(k)}(0)| \leq | P^{(k)}(0)| \) whenever \(k\) and \(m\) have opposite parity and \(0\leq k<m\); (b) if \(| p(y_j)| \leq\phi(y_j)^m\) for \(j=1,\dots,m-1\) and if \(\limsup_{y\to\infty}| p(y)| /\phi(y)^m\leq 1\), then \(| p^{(k)}(0)| \leq | Q^{(k)}(0)| \) whenever \(k\) and \(m\) have the same parity and \(0\leq k\leq m\). As was shown by the author in a previous paper, the problem of computing the extremal polynomials \(P\) and \(Q\) has applications to a Markov inequality for homogeneous polynomials on normed linear spaces.