Based on Nyquist sampling theorem, sampling would have a convolution in frequency domain, and reasonably the bandwidth of each convolution would be as that of the original signal. Nevertheless, the discrete time Fourier transform of a signal ,which has been downsampled , has the $M$ multiples bandwidth of each convolution.($M$ is the downsample factor). I could not have got the idea, what is the true relation between time/frequency domain with underlying discrete time Fourier transform?
Sampling gets convolution in frequency domain:
Downsampling gets more wider bandwidth: