International Journal of Advanced Multidisciplinary Application

A Structural Analysis of FMRA in L2(R)n

Author(s):Bijumon Ramalayathil, Haseena C, Priyanka P, Jimly Manuel

Affiliation: Department of Mathematics , Mahatma Gandhi College, Iritty, Keezhur P.O., Kerala, India.

Page No: 10-13

Volume issue & Publishing Year: Volume: 2, Issue: 10, Oct 2025

published on: 2025/10/30

Journal: International Journal of Advanced Multidisciplinary Application.(IJAMA)

ISSN NO: 3048-9350

DOI: https://doi.org/10.5281/zenodo.17338877

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Abstract:
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 establish that 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 Tkϕ1 ⊕ · · · ⊕ 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

Keywords: Frame Multiresolution Analysis (FMRA), superspace frames, dilation and translation invariance.AMS Subject Classification: 42C40, 42C15.

Reference:

• [1] S. Mallat, A Theory of Multiresolution Approximations and Wavelet Orthornomral Basis of L2(R), Tran. Amer. Math. Soc., 315 69-87 (1989).
• [2] D. Han and D. Larson, Frames, bases and group representations, Memoirs of the AMS, Volume 147, Number 697, (2000).
• [3] D. E. Dutkay, Positive definite maps, representations and frames, Reviews in Mathematical Physics, 164 (2004), 1-27.
• [4] John J. Benedetto and Oliver M. Treiber, Wavelet Frames: Multiresolution Analysis and Extension Principle, Birakauhser, Wavelet Transforms and Time- Frequency Signal Analysis -ed. Lokenath Debnath Pages Nos.3-36.
• [5] E. Hernandez and G. Weiss, A First Course on Wavelets, CRC Press, Inc., 1996, (2000).
• [6] Stefan Bildea and Dorin Ervin Dutkay and Gabriel Picioroaga, MRA Super- Wavelets, New York Journal of Mathematics, 11:1-19, (2005).
• [7] D. E. Dutkay, Positive definite maps, representations and frames, Reviews in Mathematical Physics, 164 (2004), 1-27.
• [8] A. Cohen, Ondelettes, analyses multiresolutions et traitement numerique du signal, Ph.D. Thesis, Universite Paris, Dauphine.
• Jayakumar Ramanathan, Methods of Applied Fourier Analysis , Birkhauser, Boston