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Nov 5 at 18:40 comment added Sam $n$ is tending to infinity. If what you are saying is true for every $n$, then due to smoothness of $g$, it would be a zero function @gerw. Moreover, $f$ can't be a polynomial due to constraint, $f$ is not $C^{m+1}[-1,1]$, it's regular upto $C^m$ only.
Nov 5 at 13:47 history answered gerw CC BY-SA 4.0