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Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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If the convolution of two functions $f\star g$ is equal to $g$, $f$ is even with compact sup...

The sought-after statement is wrong: $g$ can be non-constant. Fourier transforming your conditions we see that the Fourier transform $\hat{g}$ is supported at points where $\hat{f}$ is equal to $1$. …
Bruno Le Floch's user avatar