Multiresolution approximation of the vector fields on T^3

TL;DR

This paper develops a multiresolution approximation framework for vector fields on the three-dimensional torus T^3, introducing helical vectors for orthogonal decomposition and analyzing the properties of solenoidal wavelets.

Contribution

It introduces a novel helical vector basis in Fourier space and constructs a divergence-free orthonormal basis for MRA of vector fields on T^3, linking Hodge and Beltrami decompositions.

Findings

Established orthogonal decomposition of L^2(T^3) using helical vectors.

Presented a general procedure for constructing divergence-free bases from scalar bases.

Numerically investigated the localization and structure of solenoidal wavelets.

Abstract

Multiresolution approximation (MRA) of the vector fields on T^3 is studied. We introduced in the Fourier space a triad of vector fields called helical vectors which derived from the spherical coordinate system basis. Utilizing the helical vectors, we proved the orthogonal decomposition of L^2(T^3) which is a synthesis of the Hodge decomposition of the differential 1- or 2-form on T^3 and the Beltrami decomposition that decompose the space of solenoidal vector fields into the eigenspaces of curl operator. In the course of proof, a general construction procedure of the divergence-free orthonormal complete basis from the basis of scalar function space is presented. Applying this procedure to MRA of L^2(T^3), we discussed the MRA of vector fields on T^3 and the analyticity and regularity of vector wavelets. It is conjectured that the solenoidal wavelet basis must break r-regular condition, i.e. some wavelet functions cannot be rapidly decreasing function because of the inevitable singularities of helical vectors. The localization property and spatial structure of solenoidal wavelets derived from the Littlewood-Paley type MRA (Meyer's wavelet) are also investigated numerically.