This paper deals with near-best approximation of a given bivariate function using elements of quarkonial tensor frames. For that purpose we apply anisotropic tensor products of the univariate B-spline quarklets introduced around 2017 by Dahlke, Keding and Raasch. We introduce the concept of bivariate quarklet trees and develop an adaptive algorithm which allows for generalized hp-approximation of a given bivariate function by selected frame elements. It is proved that this algorithm is near-best, which means that as long as some standard conditions concerning local errors are fulfilled it provides an approximation with an error close to that one of the best possible quarklet tree approximation. For this algorithm the complexity is investigated. Moreover, we use our techniques to approximate a bivariate test function with inverse-exponential rates of convergence. It can be expected that the results presented in this paper serve as important building block for the design of adaptive wavelet-hp-methods for solving PDEs in the bivariate setting with very good convergence properties.

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