I want to find an infinitely differentiable function such that $f^{(k)}(0) = a_k$, where $a_k$ is some sequence of real numbers. My attempt was to raise $e$ to some power, but I have not gotten far. I feel like there should be a pretty simple simple solution, but for some reason I can't find it.
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Just control the $C^k$ norm at each stage iteratively by controlling the interval and the bump function decay rates. – user10354138 Oct 29 '23 at 07:21