Fourier extension is an approximation method that alleviates the periodicity requirements of Fourier series and avoids the Gibbs phenomenon when approximating functions. We describe a similar extension approach using regular wavelet bases on a hypercube to approximate functions on subsets of that cube. These subsets may have a general shape. This construction is inherently associated with redundancy which leads to severe ill-conditioning, but recent theory shows that nevertheless high accuracy and numerical stability can be achieved using regularization and oversampling. Regularized least squares solvers, such as the truncated singular value decomposition, that are suited to solve the resulting ill-conditioned and skinny linear system generally have cubic computational cost. We compare several algorithms that improve on this complexity. The improvements benefit from the sparsity in and the structure of the discrete wavelet transform. We present a method that requires $\mathcal O(N)$ operations in 1-D and $\mathcal O(N^{3(d-1)/d})$ in $d$-D, $d>1$. We experimentally show that direct sparse QR solvers appear to be more time-efficient, but yield larger expansion coefficients.

中文翻译：

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使用笛卡尔网格上的小波在一般有界域上的有效函数逼近

傅里叶扩展是一种近似方法，它减轻了傅里叶级数的周期性要求，避免了逼近函数时的吉布斯现象。我们描述了一种类似的扩展方法，使用基于超立方体的规则小波来逼近该立方体的子集上的函数。这些子集可以具有一般形状。这种结构本质上与导致严重病态的冗余相关联，但最近的理论表明，使用正则化和过采样仍然可以实现高精度和数值稳定性。正则化最小二乘求解器，例如截断奇异值分解，适用于求解由此产生的病态和瘦线性系统，通常具有三次计算成本。我们比较了几种改进这种复杂性的算法。改进受益于离散小波变换的稀疏性和结构。我们提出了一种方法，它需要一维中的 $\mathcal O(N)$ 操作和 $d$-D 中的 $\mathcal O(N^{3(d-1)/d})$，$d>1 $. 我们通过实验表明，直接稀疏 QR 求解器似乎更省时，但会产生更大的膨胀系数。