Internal structure of the multiresolution analyses defined by the unitary extension principle

We analyze the internal structure of the multiresolution analyses of L2(Rd)L2(Rd) defined by the unitary extension principle (UEP) of Ron and Shen. Suppose we have a wavelet tight frame defined by the UEP. Define V0V0 to be the closed linear span of the shifts of the scaling function and W0W0 that of the shifts of the wavelets. Finally, define V1V1 to be the dyadic dilation of V0V0. We characterize the conditions that V1=W0V1=W0, that V1=V0∔W0V1=V0∔W0 and V1=V0⊕W0V1=V0⊕W0. In particular, we show that if we construct a wavelet frame of L2(R)L2(R) from the UEP by using two trigonometric filters, then V1=V0∔W0V1=V0∔W0; and show that V1=W0V1=W0 for the BB-spline example of Ron and Shen. A more detailed analysis of the various ‘wavelet spaces’ defined by the BB-spline example of Ron and Shen is also included.