Nonlinear expansions in Banach spaces

We introduce an expansion scheme in Banach spaces, which is a generalization of the known expansion method for Reproducing Kernel Hilbert Spaces (RKHS). In particular, as an important special case, our scheme includes the well-known Blaschke unwinding series expansion in Hardy spaces \(H^p\) $H^p$ , \(1<p<\infty \) $1<p<\infty$ . Moreover, by incorporating the concept of uniqueness sets, the new method is even more general in the context of Hilbert spaces. Following the dichotomy phenomenon observed in RKHS, a similar dichotomy arises in this setting, which is derived from the local behavior of multipliers utilized in the expansion scheme.