On filter banks in spline wavelet transform on a non-uniform grid

An explicit representation of filter banks for constructing the wavelet transform of spaces of linear minimal splines on non-uniform grids on a segment is obtained. The decomposition and reconstruction operators are constructed, their mutual inverse is proved. The relations connecting the corresponding filters are established. The approach to constructing the spline wavelet decompositions used in this paper is based on approximation relations as the initial structure for constructing spaces of minimal splines and calibration relations to prove the embedding of the corresponding spaces. The advantages of the approach proposed, due to rejecting the formalism of the Hilbert spaces, are in the possibility of using non-uniform grids and fairly arbitrary non-polynomial spline wavelets.