Interpolation and differentiation for periodic functions

Summary

The two matricesX andD, of ordern, defined, in terms ofn arbitrary numbersx _j (different from each other mod(L)), by the formulaeX _jk= δ _xj ,$D_{jk} = (\pi /L)\sum\limits_{m = 1}^n {\prime cotg[(\pi /L)(x_j - x_m )]} if j = k, D_{jk} = (\pi /$, provide a convenient finite-dimensional representation of the (multiplicative) operatorx and of the corresponding differential operator d/dx, projected into an appropriate sub-space of the functional space of periodic (or antiperiodic) functions ofx, of periodL. Some implications of this finding are outlined.