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I wonder if it is possible to find a countable set $\{x_n\}$ so that certain infinite linear combinations of functions $e^{(itx_n)}$ will give functions $t^k$. If this is possible, then coefficients will have to form unbounded sets, and clearly we have to choose $\{x_n\}$ so that zeros of $\cos(tx_n)$ or of $\sin(tx_n)$ do not coincide. If this is possible, then we can use Taylor series to construct any $e^{(itx)}$.

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    $\begingroup$ You need to specify the domain of these functions, and also if you want to talk about infinite linear combinations, you need to specify in which sense you want the infinite sum to converge. Also the question in the title appears to be different from what you are asking in the actual question. $\endgroup$ – Lorenzo Quarisa Aug 26 '18 at 16:17

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