Does there exist a power series $\sum_i a_i x^i$ that is $1$ at $0$ and $0$ at integers from $1$ to $n$, and such that $\sum_i |a_i|$ is polynomial in $n$?
I feel the answer might be no but I'm not sure how to prove it.
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$\begingroup$ oh wait, the sum of absolute values of the $a_i$'s in your example grows polynomially in $n$ no? This is what I was asking for $\endgroup$– alesiaCommented Mar 29, 2020 at 16:20
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1 Answer
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Take an entire function $f$ such that $f(0)=1$ and $f(j) = 0$ for all nonzero integers: an example is $f(z) = \sin(\pi z)/(\pi z) $ for $z \ne 0$, $1$ for $z=0$.
The Maclaurin series of $f$ satisfies $\sum_{i} |a_i| < \infty$.
answered Mar 29, 2020 at 16:49