We find a close relationship between the discrete Karhunen-Loeve transform (KLT) and the discrete prolate spheroidal wave functions (DPSWF). We show that the DPSWF form a natural basis for an expansion of the eigenfunctions of the KLT in the frequency domain, and then determine more general conditions that any set of functions must obey to be a valid basis. We also present approximate solutions for small, medium, and large filter orders. The medium order solution suggests that the principal eigenfunction is, to a high degree of approximation, the principal DPSWF modulated so that its center frequency coincides with the peak of maximum energy in the signal spectrum. We then use this result to propose a new basis.