"the interval" The unit interval [0,1]
"interval" A nonzero sub-interval of [0,1], possibly all of it ("sub-interval") Always dyadic in our case, of the form (i/2^n, (i+1)/2^n)
"interval product" A cartesian product of D sub-intervals in D dimensions
"resolution" An interval of the form (i/2^n, (i+1)/2^n) has resolution n When n = 0 this involves functions that can be nonzero on all of [0,1] In order to count hat basis functions that do not vanish on the boundaries we label them with resolution(s) '-1' (and '-2' for hat basis) for simplicity In the Galerkin basis, resolution '-1' corresponds to Legendre polynomials on [0,1]
"level" The level of resolution. This is just n above, when n >= 0, and
"cell" For a given level l of resolution, a specifier for one of the above sub-intervals
"mode" In the Galerkin Scheme of degree k, each (level, cell) specifies a sub-interval on which we have k basis functions (polynomials up to degree p = k-1). mode uniquely determines one of the k such basis functions
"multi-level" A vector of D numbers specifying the resolution along each of D dimensions variable name: "levels"
"multi-cell" A vector of D numbers specifying a specific interval product given a multilevel variable name: "cells"
"multi-mode" A vector of D numbers specifying a specific product of galerkin basis functions for variable name: "modes" each interval associated with the interval product obtained from (levels, cells)
"block" The space of basis functions associated to a multilevel