The other day I was thinking of power series expansions of functions $x\to f(x)$ of the like which are defined by minimizing a Schatten $p$ integral-norm:
$$p(x) = \sum_{k=0}^{\infty} c_kx^k : \min_{\{c_k\}}\left(\int_{a}^{b}|f(x)-p(x)|^pdx\right)^{1/p}$$
What relations in terms of properties to Taylor series will they have?
Some own work:
I do expect we must get convergence at least on $x\in [a,b]$ or else the above would not be defined (?) I do not know how to prove this.
I also suspect that any limit as $b\to a$, we should be getting the Taylor expansion around $a$, but even this I don't know how to prove.