We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low dimensional variable interactions. Compactly supported periodic Chui-Wang wavelets are used for the tensorized hyperbolic wavelet basis. In a first step we give a self-contained characterization of tensor product Sobolev-Besov spaces on the $d$-torus with arbitrary smoothness in terms of the decay of such wavelet coefficients. In the second part we perform and analyze scattered-data approximation using a hyperbolic cross type truncation of the basis expansion for the associated least squares method. The corresponding system matrix is sparse due to the compact support of the wavelets, which leads to a significant acceleration of the matrix vector multiplication. In case of i.i.d. samples we can even bound the approximation error with high probability by loosing only $log$-terms that do not depend on $d$ compared to the best approximation. In addition, if the function has low effective dimension (i.e. only interactions of few variables), we qualitatively determine the variable interactions and omit ANOVA terms with low variance in a second step in order to increase the accuracy. This allows us to suggest an adapted model for the approximation. Numerical results show the efficiency of the proposed method.
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