The wandering subspace property and Shimorin’s condition of shift operator on the weighted Bergman spaces

In the present paper, we first study the wandering subspace property of the shift operator on the $$I_{a}$$ I a type zero based invariant subspaces of the weighted Bergman spaces $$L_{a}^{2}(dA_{n})(n=0,2)$$ L a 2 ( d A n ) ( n = 0 , 2 ) via the spectrum of some Toeplitz operators on the Hardy space $$H^{2}$$ H 2 . Second, we give examples to show that Shimorin’s condition for the shift operator fails on the $$I_{a}$$ I a type zero based invariant subspaces of the weighted Bergman spaces $$L_{a}^{2}(dA_{\alpha })(\alpha >0)$$ L a 2 ( d A α ) ( α > 0 ) .

TIME-SHIFTS GENERALIZED MULTIRESOLUTION ANALYSIS OVER DYADIC-SCALING REDUCING SUBSPACES

A Generalized Multiresolution Analysis (GMRA) associated with a wavelet is a sequence of nested subspaces of the function space ℒ2(ℝ), with specific properties, and arranged in such a way that each of the subspaces corresponds to a scale 2m over all time-shifts n. These subspaces can be expressed in terms of a generating-wandering subspace — of the dyadic-scaling operator — spanned by orthonormal wavelet-functions — generated from the wavelet. In this paper we show that a GMRA can also be expressed in terms of subspaces for each time-shift n over all scales 2m. This is achieved by means of "elementary" reducing subspaces of the dyadic-scaling operator. Consequently, Time-Shifts GMRA associated with wavelets, as well as "sub-GMRA" associated with "sub-wavelets" will then be introduced.