Convolution Operations Arising From Vandermonde Matrices

Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution, and include additive and multiplicative convolution of Vandermonde matrices and deterministic diagonal matrices, and cases where two independent Vandermonde matrices are involved. It is also shown that the convergence of any combination of Vandermonde matrices is almost sure. The convolutions are divided into two types: those which depend on the phase distribution of the Vandermonde matrices, and those which depend only on the spectra of the matrices. A general criterion is presented to find which type applies for any given convolution. A simulation is presented, verifying the results. Implementations of the presented convolutions are provided and discussed. The implementation is based on the technique of Fourier-Motzkin elimination, and is quite general as it can be applied to virtually any combination of Vandermonde matrices. Connections with related matrices, such as Toeplitz and Hankel matrices, are also discussed.