Question about convolution in time and frequency domain
Normally, the variables $t$ and $f$ are used for continuous time and frequency. But from your question I understand that you're talking about discrete time and frequency, and their relation via the Discrete Fourier Transform (DFT).

If you have discrete-time sequences $a[n]$ and $b[n]$ and their DFTs $A[k]$ and $B[k]$, then the DFT of the linear convolution $c[n]=a[n]\star b[n]$ is NOT equal to $A[k]\cdot B[k]$. The product of the DFTs corresponds to circular (or cyclic) convolution in the time domain.

The linear convolution of two $16$-point sequences has indeed $2\cdot 16-1=31$ points, whereas the circular convolution of two $16$-point sequences also has $16$ points.