This chapter establishes the exponential decay of gamblets under an appropriate notion of distance derived from subspace decompositionin a way that generalizesdomain decomposition in the computation of PDEs.The first stepspresent sufficient conditions forlocalizationbased on a generalization of the Schwarz subspace decomposition and iterative correction methodintroduced by Kornhuber and Yserentantand the LOD method of Malqvist and Peterseim. However,when equipped withnonconforming measurement functions, one cannot directly work in the primal space, but instead one has to find ways to work in the dual space. Therefore, the next steps presentnecessary and sufficient conditions expressed as frame inequalities in dual spaces that, in applications to linear operators on Sobolev spaces,are expressed as Poincaré, inverse Poincaré, and frame inequalities.

This chapter reviews classical homogenizationconcepts such as the cell problem; correctors; compactness by compensation; oscillating test functions; H, G, and Gamma convergence; and periodic and stochastic homogenization. Numerical homogenization is presented as the problem of identifying basis functions that are both as accurate and as localized as possible. Optimal recovery splines constructed from simple measurement functions (Diracs, indicator functions, and local polynomials) provide a simple to solution to this problem: they achieve the Kolmogorov n-width optimal accuracy (up to a constant) and they are exponentially localized. Current numerical homogenization methods are reviewed. Gamblets, the LOD method, the variational multiscale method, andpolyharmonic splines are shown to have a common characterization as optimal recovery splines.