This chapter generalizes and extends the development ofoperator-adapted wavelets (gamblets)and their resulting multiresolution decompositionsfrom Sobolev spaces to Banach spaces equipped with a quadraticnorm and a nonstandard dual pairing. The fundamental importance of the Schur complement is elucidated and the geometric nature of gamblets is presented from two views: one regarding basis transformations derived from the nesting, and the other the linear transformations associated with these basis transformations. A table of gamblet identities is presented.

By representing the operator as independent, sparse, well-conditioned linear systems, theoperator-adapted wavelet (gamblet) transform of the previous chapter naturally leads to a scalable linear solver with some degree of universality. The near-linear complexity of the solver and the Fast Gamblet Transform are based on the nesting, exponential localization, and Riesz stability of the underlying wavelets. The representation of the Green's function in the basis formed by these wavelets is sparse and rank-revealing.The algorithm isillustrated through a numerical application toa second-order divergence form elliptic operator with rough coefficients.

The computation ofgamblets is accelerated by localizing their computation in a hierarchical manner (using a hierarchy of distances), and the approximation errors caused by these localization steps are bounded based on three properties: nesting, the well-conditioned nature of the linear systems solved in the Gamblet Transform, and theexponential decay of the gamblets. These efficiently computed, accurate, andlocalized gamblets are shown to producea Fast Gamblet Transform of near-linear complexity. Application to the three primary classes ofmeasurement functions in Sobolev spaces are developed.