Using the refinement equations for the construction of prewavelets. II: Powers of two. (English) Zbl 0777.41013
Curves and surfaces, Pap. Int. Conf., Chamonix-Mont-Blanc/Fr. 1990, 209-246 (1991).
Summary: [For the entire collection see Zbl 0729.00010.]
[For part I see the second author, Numer. Algorithms 1, No. 1, 75-116 (1991; Zbl 0759.65005).]
We study basic questions of wavelet decompositions associated with multiresolution analysis. A rather complete analysis of multiresolutions associated with the solution of a refinement equation is presented. The notion of extensibility of a finite set of Laurent polynomials is shown to be central in the construction of wavelets by decomposition of spaces. Two examples of extensibility, first over the torus and then in complex space minus the coordinate axes are discussed. In each case we are led to a decomposition of the fine space in a multiresolution analysis as a sum of the adjacent coarse space plus an additional space spanned by the multi-integer translates of a finite number of pre-wavelets. Several examples are provided throughout to illustrate the general theory.
[For part I see the second author, Numer. Algorithms 1, No. 1, 75-116 (1991; Zbl 0759.65005).]
We study basic questions of wavelet decompositions associated with multiresolution analysis. A rather complete analysis of multiresolutions associated with the solution of a refinement equation is presented. The notion of extensibility of a finite set of Laurent polynomials is shown to be central in the construction of wavelets by decomposition of spaces. Two examples of extensibility, first over the torus and then in complex space minus the coordinate axes are discussed. In each case we are led to a decomposition of the fine space in a multiresolution analysis as a sum of the adjacent coarse space plus an additional space spanned by the multi-integer translates of a finite number of pre-wavelets. Several examples are provided throughout to illustrate the general theory.
MSC:
| 41A15 | Spline approximation |
| 65D20 | Computation of special functions and constants, construction of tables |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
| 41A63 | Multidimensional problems |
