We tensorize the Faber spline system from [14] to prove sequence space isomorphisms for multivariate function spaces with higher mixed regularity. The respective basis coefficients are local linear combinations of discrete function values similar as
for the classical Faber Schauder system. This allows for a sparse representation of the function using a truncated series expansion by only storing discrete (finite) set of function values. The set of nodes where the function values are taken depends on the respective function in a non-linear way. Indeed, if we choose the basis functions adaptively it requires significantly less function values to represent the initial function up to accuracy $varepsilon>0$ (say in $L_infty$) compared to hyperbolic cross projections. In addition, due to the higher regularity of the Faber splines we overcome the (mixed) smoothness restriction $r<2$ and benefit from higher mixed regularity of the function. As a byproduct we present the solution of Problem 3.13 in the Triebel monograph [46] for the multivariate setting.
This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the piecewise li
near 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 (exponentially decaying) and the main parts are concentrated on a segment. This construction gives a complete answer to Problem 3.13 in Triebels monograph (see References [47]) by extending the classical Faber basis to higher orders. Roughly, the crucial idea to obtain a higher order Faber spline basis is to apply Taylors remainder formula to the dual Chui-Wang wavelets. As a first step we explicitly determine these dual wavelets which may be of independent interest. Using this new basis we provide sampling characterizations for Besov and Triebel-Lizorkin spaces and overcome the smoothness restriction coming from the classical piecewise linear Faber-Schauder system. This basis is unconditional and coefficient functionals are computed from discrete function values similar as for the Faber-Schauder situation.
