Timeline for Uniformly approximating a function and its derivative using polynomials
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 7, 2020 at 22:36 | comment | added | Pietro Majer | I don't know of the extensions to polynomials, but I guess there are some. An obvious extension is the case of polynomials $p_k$ that maybe are not monomials but whose span coincides with a span of monomials $x^{n_k}$ satisfying MS hypothesis. Or also, polynomials $p_k=L(x^{n_k})$ for $L$ an invertible operator on $C^0(K)$ | |
| Oct 7, 2020 at 22:26 | history | edited | Pietro Majer | CC BY-SA 4.0 |
edited body
|
| Oct 7, 2020 at 22:16 | comment | added | mw19930312 | Thanks for the reply! I know of M\"{u}ntz-Szasz theorem. May I ask if there are any extensions on the cases where $p_k$'s are not monomials? The cases that I'm mostly interested in are (1) $p_k = a^k b$ for some fixed polynomials $a$ and $b$ (2) $p_k$'s are vector-valued polynomials. But these cases seem to be more complicated and less-addressed in the literature of polynomial approximation. | |
| Oct 7, 2020 at 21:57 | comment | added | Pietro Majer | Note I assumed $0\notin K$, as it is needed to apply the Muntz-Szász Theorem. If $0\in K$, the trivial exceptions can be ruled out e.g. assuming $ n_0=0$ and $n_1=1 $ . | |
| Oct 7, 2020 at 21:31 | history | answered | Pietro Majer | CC BY-SA 4.0 |