The Study of Periodic Tight Framelets and Wavelet Frame Packets and Applications

Abstract

Information science focuses on understanding problems from the perspective of the stake holders involved and then applying information and other technologies as needed. A necessary and sufficient condition is identified in term of refinement masks for applying the unitary extension principle for periodic functions to construct tight wavelet frames. Then a theory on the approximation order of truncated tight frame series is established, which facilitates construction of tight periodic wavelet frames with desirable approximation order. The pyramid decomposition scheme is derived based on the generalized multiresolution structure. Introduction and Concepts The setup of tight wavelet frames provides great flexibility in approximating and representing periodic functions. Fundamentals issues involved include the construction of tight periodic wavelet frames, approximation powers of such wavelet frames, and whether wavelet frames lead to sparse representat of locally smooth periodic functions. The frame theory plays an important role in the modern time-frequency analysis. It has been developed very fast over the last twenty years, espe cially in the context of wavelets and Gabor systems. This scientific field investigates the relationship betwenrse represeen the structure of materials at atomic or molecular scales and their macroscopic properties. Wavelet theory has been applied to signal processing, image compression, and so on. Frames for a separable Hilbert space were formally defined by Duffin and Schaeffer [1] in 1952 to study some deep problems in nonharmonic Fourier series. Basically, Duffin and Schaeffer abstracted the fundamental notion of Gabor for studying signal processing [2]. These ideas did not seem to generate much general interest outside of nonharmonic Fourier series however (see Young's [3]) until the landmark paper of Daubechies, Grossmann, and Meyer [4] in 1986. After this groundbreaking work, the theory of frames began to be more widely studied both in theory and in applications [5,6], such as signal processing, image processing, data compression, sampling theory. We begin with the unitary extension pinciple and formulate a general procedure for constructing wavelet frames. The emphasis is on having refinement masks as the starting point. The condition for this can be easily verified and also provide insight to the refinement masks that enable the construction process. Let [0, 2 ] L a be the space of all 2a -periodic square-integrable complex-valued func tions over the real line R with inner product , given by 2 0 , 1/(2 ) ( ) ( ) a h v a h x v x dx = ∫ where 2 ( ), ( ) [0, 2 ] h x v x L a ∈ , and norm 1/ 2 2 || || , u u u = . For a function 2 ( ) [0, 2 ] h x L a ∈ , we denote its Fourier series as ( ) in n Z n e h ∈ ∑ , where ( ) , in n h e h = , n Z ∈ , are its Fourier coefficient. For any 0 , u Z ≤ ∈ we define the shift operator 2 2 [0, 2 ] [0, 2 ] : u S L a L a  S by ( ) : ( 2 / 2 ) u u S h x h x a = −   . For 2 ( ) [0, 2 ] h x L a ∈ , since ( ) h x is a periodic function, it suffices to consider the shifts ( ) u k S h x ∈Λ   , , where k Λ is given by International Conference on Materials Engineering and Information Technology Applications (MEITA 2015) © 2015. The authors Published by Atlantis Press 240 1 1 1 1 { 2 1, 2 2, , 2 1, 2 } k k k k k − − − − Λ = − + − + −  : . Let (2 ) k ∆ be the 2 − periodic complex sequences k b , that is k k k b u b   ( +2 )= ( ) for all ,u Z ∈  . We denote the discrete Fourier transform of (2 ) k k b ∈∆ by 2 /2 ( ) : ( ) k k ai n k k b n b e − ∈Ω =∑ 