I want to compute the $K$ sums: $$ f_k = \sum_{j=0}^{N-1} a_j e^{i \frac{2 \pi j k}{K}} $$ for $f_0,f_1,...f_{K-1}$. Now if $K = N$, then this is the usual DFT, but in my application $K \ne N$, as I have a different number of coefficients $a_j$ than grid points $k$. In other words the DFT matrix here is not square.
Now if $K > N$ it seems that I can expand the sum to $$ f_k = \sum_{j=0}^{K-1} a_j e^{i \frac{2 \pi j k}{K}} $$ and set $a_N,...,a_{K-1} = 0$ (i.e., zero-pad the coefficients $a_j$ to make the DFT matrix square). But this is inefficient since it will add unnecessary operations to the FFT by adding zero terms to the sums.
Also, if $N > K$ I don't see an easy way to turn this into a standard DFT problem. Does anyone know of any tricks?
