Nonstationary Multiwavelets and Multiwavelet Packets in Sobolev Space $$H^s({\mathbb {R}}^d)$$

Abstract

Nonstationary multiresolution analysis has different scaling and wavelet functions at different scales. In Bastin and Laubin (Duke Math J 87:481–508, 1997), nonstationary wavelets in Sobolev spaces were introduced. Here we construct a nonstationary multiresolution analysis of multiwavelets for the higher-dimensional Sobolev spaces. We give the construction of nonstationary multiwavelets and derived their orthogonal properties in Sobolev space $$H^s({\mathbb {R}}^d)$$ . We perform some splitting trick over nonstationary multiwavelets and multiwavelet spaces to construct nonstationary multiwavelet packet and multiwavelet packet spaces in Sobolev space $$H^s({\mathbb {R}}^d)$$ . A nonstationary multiwavelet and multiwavelet packet expansion of Bessel potentials is given as an application of an orthonormal multiwavelet and multiwavelet packet basis in Sobolev space.