The Projection Method in the Development of Wavelet Bases

Most of the research work on wavelet analysis so far has been concentrated on wavelets on uniform meshes in Euclidean spaces. We are interested in wavelet bases for function spaces on bounded domains with possibly nonuniform or irregular meshes. For this purpose, we introduce the projection method for construction of wavelet bases. Let (Vn)n=0,1,2,... be a family of closed subspaces of a Hilbert space H. Suppose that V0 = {0} and Vn ⊂ Vn+1 for n = 0, 1, . . . . Let Pn be a linear projection from Vn+1 onto Vn. If Wn denotes the kernel space of Pn, then Vn+1 is the direct sum of Vn and Wn. We give necessary and sufficient conditions on the projections Pn (n = 0, 1, . . . ) such that the combination of Riesz bases of Wn (n = 0, 1, . . . ) forms a Riesz basis of H. Under the guidance of the general theory, we introduce discrete wavelets on intervals and discuss their applications to image processing. We also investigate wavelet bases of splines for Sobolev spaces on bounded domains and their applications to numerical solutions of partial differential equations. 2000 Mathematics Subject Classification: 42C40, 41A15, 46B15, 46E35.