show that the Fourier transform is a linear transform, that is F[ax(t) + by(t)] = aF[x(t)] + bF[y(t)]. Prove the convolution theorm for the Fourier transform, which states that convolution in the time...

show that the Fourier transform is a linear transform, that is F[ax(t) + by(t)] = aF[x(t)] + bF[y(t)]. Prove the convolution theorm for the Fourier transform, which states that convolution in the time domain corresponds to multiplication in the frequency domain: F[x(t) ∗ y(t)] = X(jω)Y (jω), where X(jω) = F[x(t)] and Y (jω) = F[y(t)].