A higher order Faber spline basis for nonlinear approximation of functions with mixed smoothness

Our goal in this talk is twofold. On the one hand, we present the construction of a higher order Faber spline basis and application of this basis to sampling discretization of functions of mixed smoothness with higher regularity. This new basis has similar properties as the piecewise linear classical Faber-Schauder basis except for the compactness of the support. Although the new basis functions are supported on the real line they are very well localized (exponential decay) and the main parts are concentrated on a segment. The multivariate basis is constructed by using tensor product approach. This construction gives a complete answer to Problem 3.13 in Triebel's monograph (Faber systems and their use in sampling, discrepancy, numerical integration, 2012) by extending the classical Faber basis to higher orders. Using this new basis we provide sampling characterizations for Besov spaces of mixed smoothness and overcome the smoothness restriction coming from the classical piecewise linear Faber-Schauder system. On the other hand, we apply the obtained characterization for finding the upper order estimates for best n-term approximation with respect to this Faber spline basis. The main advantage of this approach is that the aggregate of best n-term approximation is constructed by using only function values.