Let $f(x)$ be an arbitrary reasonably behaved function possessing a Fourier transform, and let its value be $1$ between $-0.5$ and $0.5$. If I multiply this function by a top hat function, $g$, where $g$ is $1$ between minus half and plus half, I get a top hat function, as a result. Multiplication in real domain should be equivalent to convolution in Fourier domain. So the convolution of $FT(f)$ and $FT(g)$ should give a sinc function, which is the FT of a top hat. $FT(g)$ is also sinc function (since it's also FT of a top-hat). So if I convolve $FT(f)$ with a sinc, I should always get a sinc.
Is the above reasoning correct? The result does not seem reasonable. Where am I wrong?