Homogeneous Dirichlet wavelets on the interval diagonalizing the derivative operator, and application to free-slip divergence-free wavelets

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 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.