Polynomial inequalities in Banach spaces (Q2791812)

This paper is concerned with a generalization of the classical Szegő and Bernstein-Markov inequalities to the case of polynomial mappings between Banach spaces. One of the main results is a generalization of the following Harris inequality for a Hilbert space \(X\): NEWLINE\[NEWLINE |d_xQ(v)|\leq (\deg Q)\left(\| v\|^2+\frac{(x\cdot v)^2}{1-\| x\|^2}\right)^{1/2}\| Q\|, \quad x\in\mathrm{int}(B), \quad x\in X, NEWLINE\]NEWLINE where \(B\) is the unit closed ball in \(X\) and \(x\cdot v\) denotes the scalar product of \(x\) and \(v\). In the case \(X\) is an arbitrary real Banach space, the scalar product is replaced by some pseudo inner product that is defined by using notion of the injective norm. This norm is a norm for the so called injective complexification of a real Banach space.NEWLINENEWLINEFor the entire collection see [Zbl 1337.41001].