Fourier transform or expression for $n$-th root of $sin(x)$ as $n \rightarrow \infty$.

Question

This morning I asked a question about functional square root. On the wiki page of it, I found it interesting that the $n$-th root of $sin(x)$ looks like a triangle wave as $n$ goes to infinity. Is there a special Fourier transform of $\lim\limits_{n \rightarrow \infty} \sqrt[n]{sin(x)}$, or a nice way to formulate this wave?

Answer 1

Yes, it itself (green), over half a period, is the Triangular, hat, or tent function, a convolution of two unit rectangular (boxcar) functions, and thus the Fourier transform of the square of the Sinc , normalized cardinal sine, function.

Signal processing engineers prefer its cousin, the Sawtooth wave.