Defining as monomials $m(x,n)\,:=\,x^n,\,n\in\mathbb{N}_0$, I denote by an "super-monomial" an analytic function of the form
$$ \overline{m}(x,n,(a))\ :=\ x^n+\sum\limits_{i=1}^\infty \frac{a_{n+i}x^{n+i}}{(n+i)!};\ \frac{d^{n+k}\overline{m}(x,n,(a))}{dx^{n+k}}\not\equiv 0,\, \frac{\left|a_{n+k}\right|}{(n+k)!}\le n+k\ \ \forall k\ge 0$$
Question:
- what can be said about the error term of interpolating a function $g(x)$ by a linear combination $$\sum\limits_{i=0}^{n}\alpha_i\overline{m}(x,i,(a)_i)$$ over the common interval of convergence at $n+1$ equidistant arguments; when compared to polynomial interpoation; will it be generally better or worse?
- is it possible to determine optimal sets $\lbrace(a)_i\,|\,0\le i\le n\rbrace$ of coefficient sequences that are optimal for interpolating functions with a given degree of smoothness?
- is there already an established name for what I called super-monomials?
