Local wavelet basis for an irregular grid

The spaces of $\mathcal B_\varphi$-splines are proved to be embedded for an arbitrary grid refinement; the direct (wavelet) decomposition for chains of embedded spaces of $\mathcal B_\varphi$-splines on a sequence of refined irregular grids is discussed; a wavelet basis of functions with compact supports is constructed; formulas of decomposition and reconstruction are provided. Simple solutions of certain interpolation problems in the spaces considered are suggested. Examples of the spline spaces are presented.