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.Comment: LaTeX, 33 Pages, 3 figures. submitted to J. Math. Phy
