Some Applications of Projection Operators in Wavelets

By rewriting the projection operator P 0 in wavelets in another formula, we obtain a characterization of dimJ V 0 (x) where V 0 is a ?-shift-invariant subspace of L 2 (R n) derived from a dual wavelet basis and prove that there does not exist a wavelet function 2 L 2 (R) such that ^ has compact support and k2Z Z (supp ^ + 4k) = R up to a zero subset of R. x1. Deenitions and Main Results In I.Daubechies 4], the author put forth such a question that if one imposes some smoothness conditions on the Fourier transform of the wavelet function 2 L 2 (R); does must be derived from an MRA? In 6] and 7], P.G.Lemari e gave a satisfactory positive solution to this question. The main ingredient in the recipe of his proof is a new formula of the projection operator P 0. In this paper, we shall again obtain this formula under much weaker conditions and give a complete description of dimJ V 0 (x), which means that we have a criterion for a dual wavelet basis which can be generated by an MRA. More interesting is that whether a wavelet basis can be generated by an MRA with one scaling function is totally determined by the support of the Fourier transforms of their wavelet functions. So it seems that Daubechies' question whether can be drived from an MRA is not related to the smoothness of ^ but the support of ^. In order to state our results more clearly, we shall recall some notations and concepts in advance.