A Structural Analysis of FMRA in L 2 (R)n

Frame Multiresolution Analysis (FMRA) extends the classical concept of multiresolution analysis to the context of frames in Hilbert spaces. In this paper, we investigate FMRA in the superspace L2 (R)n and establishthat the frame property of the system {Tkϕ1 ⊕ · · · ⊕ ϕn : k ∈ Z} in V0 is preserved under dilations by the operator UC. Specifically, we prove that the dilated-translated system {Uj T kϕ1 ⊕ · · · ⊕ ϕn : k ∈ Z} forms a frame for Vj with the same frame bounds as in V0. This result demonstrates the stability of FMRA frames under generalized dilation and translation operators in L2 (R)n , facilitating their application to multiple signal processing and superwavelet constructions.