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

Abstract

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.