Given cell-average data values of a piecewise-smooth bivariate function $f$ within a domain $\varOmega $, we look for a piecewise adaptive approximation to $f$. We are interested in an explicit and global (smooth) approach. Bivariate approximation techniques, as trigonometric or splines approximations, achieve reduced approximation orders near the boundary of the domain and near curves of jump singularities of the function or its derivatives. Whereas the boundary of $\varOmega $ is assumed to be known, the subdivision of $\varOmega $ to subdomains on which $f$ is smooth is unknown. The first challenge of the proposed approximation algorithm would be to find a good approximation to the curves separating the smooth subdomains of $f$. In the second stage, we simultaneously look for approximations to the different smooth segments of $f$, where on each segment we approximate the function by a linear combination of basis functions $\{p_i\}_{i=1}^M$, considering the corresponding cell averages. A discrete Laplacian operator applied to the given cell-average data intensifies the structure of the singularity of the data across the curves separating the smooth subdomains of $f$. We refer to these derived values as the signature of the data, and we use it for both approximating the singularity curves separating the different smooth regions of $f$. The main contributions here are improved convergence rates to the approximation of the singularity curves and the approximation of $f$, an explicit and global formula, and, in particular, the derivation of a piecewise-smooth high-order approximation to the function.

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来自单元平均数据的分段平滑二维函数的全局和显式逼近

给定域 $\varOmega $ 内的分段平滑二元函数 $f$ 的单元平均数据值，我们寻找 $f$ 的分段自适应逼近。我们对显式和全局（平滑）方法感兴趣。双变量逼近技术，如三角函数或样条逼近，在域边界附近和函数或其导数的跳跃奇点曲线附近实现减少的逼近阶数。虽然 $\varOmega $ 的边界是已知的，但是 $\varOmega $ 对 $f$ 平滑的子域的细分是未知的。所提出的近似算法的第一个挑战是找到一个好的近似曲线来分割$f$的平滑子域。在第二阶段，我们同时寻找 $f$ 的不同平滑段的近似值，在每个段上，我们通过基函数 $\{p_i\}_{i=1}^M$ 的线性组合来近似函数，同时考虑相应的单元平均值。应用于给定单元平均数据的离散拉普拉斯算子加强了分隔 $f$ 的平滑子域的曲线上的数据奇点结构。我们将这些派生值称为数据的签名，我们使用它来逼近分隔 $f$ 的不同平滑区域的奇异曲线。这里的主要贡献是改进了奇异曲线逼近的收敛速度和 $f$ 的逼近，这是一个明确的全局公式，特别是对函数的分段平滑高阶逼近的推导。