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

We analyze the internal structure of the multiresolution analyses of L^2(R^d) defined by the unitary extension principle (UEP) of Ron and Shen. Suppose we have a wavelet tight frame defined by the UEP. Define V"0 to be the closed linear span of the shifts of the scaling function and W"0 that of the shifts of the wavelets. Finally, define V"1 to be the dyadic dilation of V"0. We characterize the conditions that V"1=W"0, that V"1=V"0@?W"0 and V"1=V"0@?W"0. In particular, we show that if we construct a wavelet frame of L^2(R) from the UEP by using two trigonometric filters, then V"1=V"0@?W"0; and show that V"1=W"0 for the B-spline example of Ron and Shen. A more detailed analysis of the various 'wavelet spaces' defined by the B-spline example of Ron and Shen is also included.

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