The principles of dilation and shift are two important properties that are attributed to wavelets. It is shown that inclusion of such properties in the choice of a basis in Galerkin's method can lead to a slow growth of the condition number of the system matrix obtained from the discretization of the differential form of Maxwell's equations. It is shown that for one-dimensional problems the system matrix can be diagonalized. For two-dimensional problems, however, the system matrix can be made mostly diagonal. This paper illustrates the application of the new type of ''dilated'' basis for a Galerkin's method (or equivalent, for example, finite element method) for the efficient solution of waveguide problems. Typical numerical results are presented to illustrate the concepts. ¿ American Geophysical Union 1994