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Given cell-average data values of a piecewise smooth bivariate function $f$within a domain $\Omega$, we look for a piecewise adaptive approximation to$f$. We are interested in an explicit and global (smooth) approach. Bivariateapproximation techniques, as trigonometric or splines approximations, achievereduced approximation orders near the boundary of the domain and near curves ofjump singularities of the function or its derivatives. Whereas the boundary of$\Omega$ is assumed to be known, the subdivision of $\Omega$ to subdomains onwhich $f$ is smooth is unknown. The first challenge of the proposedapproximation algorithm would be to find a good approximation to the curvesseparating the smooth subdomains of $f$. In the second stage, we simultaneouslylook for approximations to the different smooth segments of $f$, where on eachsegment we approximate the function by a linear combination of basis functions$\{p_i\}_{i=1}^M$, considering the corresponding cell-averages. A discreteLaplacian operator applied to the given cell-average data intensifies thestructure of the singularity of the data across the curves separating thesmooth subdomains of $f$. We refer to these derived values as the signature ofthe data, and we use it for both approximating the singularity curvesseparating the different smooth regions of $f$. The main contributions here areimproved convergence rates to both the approximation of the singularity curvesand the approximation of $f$, an explicit and global formula, and, inparticular, the derivation of a piecewise smooth high order approximation tothe function.

Global and explicit approximation of piecewise smooth 2D functions from cell-average data