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An explicit representation of filter banks is obtained for constructing wavelet transforms of spaces of linear minimal splines on nonuniform grids on a segment. Decomposition and reconstruction operators are constructed, and it is proved that they are mutually inverse. Some interrelations between the corresponding filters are established. The approach to constructing spline wavelet decompositions used in the present paper is based on using the approximating relations as initial structures for constructing spaces of minimal splines and the calibration relations to prove that the corresponding spaces are embedded An advantage of the approach is the possibility of using nonuniform grids and arbitrary nonpolynomial spline wavelets without using the formalism of Hilbert spaces.

The paper presents a matrix representation of filter banks corresponding to spline wavelets with shifted supports. The matrix form of the decomposition and reconstruction filters simplifies the representation of nonuniform nonstationary wavelet transforms built on nonuniform grids on a finite interval. Such a representation of filters is used, for example, in constructing error-correcting codes.