This paper presents a new construction of a homogeneous Dirichlet wavelet basis on the unit interval, linked by a diagonal differentiation-integration relation to a standard biorthogonal wavelet basis. This allows to compute the solution of Poisson equation by renormalizing the wavelet coefficients - as in Fourier domain but using locally supported basis functions with boundary conditions-, which yields a linear complexity $\mathcal{O}(N)$ for this problem. Another application concerns the construction of free-slip divergence-free wavelet bases of the hypercube, in general dimension, with an associated decomposition algorithm as simple as in the periodic case.

中文翻译：

---

对角化导数算子的区间上的齐次狄利克雷小波及其在无自由滑散小波中的应用

本文提出了单位区间上齐次狄利克雷小波基的新构造，通过对角线微分积分关系与标准双正交小波基相关联。这允许通过重新归一化小波系数来计算泊松方程的解 - 就像在傅立叶域中一样，但使用具有边界条件的局部支持的基函数 - 这会产生线性复杂度$\mathcal{哦}(N)$对于这个问题。另一个应用涉及超立方体的自由滑移无散度小波基的构建，在一般维度上，关联的分解算法与周期情况一样简单。