Generalization of the polarization formula for nonhomogeneous polynomials and analytic mappings on Banach spaces (Q968900)

The authors use the generalised Rademacher functions \((s_j^{[n]})\) to obtain polarisation formulae for nonhomogeneous polynomials and analytic mappings of bounded type on complex Banach spaces. They show that if \(P\) is a polynomial of degree \(n\) on a complex Banach space \(E\) then \(A_k\), the symmetric \(k\)-linear mapping associated with the \(k\)-homogeneous part of \(P\), is given by \[ A_k(x_1,\dots,x_k)=\Pi_k(P)(x_1,\dots,x_k)+\sum_{j=1}^r(-1)^i\sum_{{\mathbb{N}}\ni p\leq r}\Pi_{pk}(P)(x_1^p,\dots,x_k^p) \] where \(r=\left[{n\over k}\right]\), \({\mathbf N}_i=\{p_1p_2\dots p_i:p_1<p_2<\dots<p_i\text{ are prime numbers}\}\) and \[ \Pi_k(P)(x_1,\dots,x_k)={1\over k!}\int_0^1(S_1^{[k]})^{k-1}(t)\dots (S_k^{[k]})^{k-1}(t)P(S_1^{[k]}(t)x_1+\cdots +S_k^{[k]}(t)x_k)\,dt. \] When \(f\) is an entire analytic mapping of bounded type it is shown that the symmetric \(k\)-linear mapping associated with, \(P_k\), the \(k\)-homogeneous part of \(f\), is given by \[ A_k(x_1,\dots,x_k)=\Pi_k(f)(x_1,\dots,x_k)+\sum_{j=1}^\infty(-1)^i \sum_{p\in {\mathbf N}}\Pi_{pk}(f)(x_1^p,\dots,x_k^p) \] where \(\Pi_k(f)(x_1,\dots,x_k)=\lim_{m\to\infty}\Pi_k(P_m)(x_1,\dots,x_k)\).