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We tensorize the Faber spline system from [14] to prove sequence spaceisomorphisms for multivariate function spaces with higher mixed regularity. Therespective basis coefficients are local linear combinations of discretefunction values similar as for the classical Faber Schauder system. This allowsfor a sparse representation of the function using a truncated series expansionby only storing discrete (finite) set of function values. The set of nodeswhere the function values are taken depends on the respective function in anon-linear way. Indeed, if we choose the basis functions adaptively it requiressignificantly less function values to represent the initial function up toaccuracy $\varepsilon>0$ (say in $L_\infty$) compared to hyperbolic crossprojections. In addition, due to the higher regularity of the Faber splines weovercome the (mixed) smoothness restriction $r<2$ and benefit from higher mixedregularity of the function. As a byproduct we present the solution of Problem3.13 in the Triebel monograph [46] for the multivariate setting.

Adaptive sampling recovery of functions with higher mixed regularity