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Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. In this paper, we discuss a new set of symmetric tight frame wave -lets with the associated filterbanks outputs downsampled by several generators. The frames consist of several generators obtained from the lowpass filter using spectral factorization, with lowpass fil -ter via a simple approach using Legendre polynomials. The filters are feasible to be designed and offer smooth scaling functions and frame wavelets. We shall give an example to demonstrste that so-me examples of symmetric tight wavelet frames with three compactly supported real-valued sym-metric generators will be presented to illustrate the results. Introduction Manufacturing engineering is an interdisciplinary field applying the properties of matter to var -ious areas of science and engineering. Mechanical engineers apply the principles of mechanics and energy to the design of machines and devices: Energy and Motion. The frame theory has been one of powerful tools for researching into wavelets. Duffin and Schaeffer introduced the notion of frames for a separable Hilbert space in 1952. Later, Daubechies etal revived the study of frames in[1,2], and since then, fra -mes have become the focus of active research, both in theory and in applications, such as signal processing, image processing and sampling theory. The rise of frame theory in applied mathematics is due to the flexibility and redundancy of frames, where robustness, error tolerance and noise suppression play a vital role [3,4]. The concept of frame multiresolution analysis (FMRA) as described in [2] generalizes the notion of MRA by allowing non-exact affine frames. However, subspaces at different resolutions in a FMRA are still generated by a frame formed by translates and dilates of a single function. This paper is motivated from the observation that standard methods in sampling theory provide examples of multiresolution structure which are not FMRAs. Inspired by [2] and [5], we introduce the notion of a generalized multiresolution structure(BGMS) of 2 2 ( ) L R , which has a pyramid decomposition scheme. It also lead to new constructions of affine frames of 2 2 ( ) L R . Fundamental Properties of Gabor Frames Let W be a separable Hilbert space and Λ is an index set. We recall that a sequence { } : v v Z W ∈ ⊆  is a frame for H if there exist positive real numbers B , C such that , W ∀ ∈  2 2 2 , v v B C ∈Λ ≤ ≤ ∑     (1) A sequence { } : v v Z W ∈ ⊆  is a Bessel sequence if (only) the upper inequality of ( ) 1 holds. If only for all W ∈W⊂  , the upper inequality of ( ) 1 holds, the sequence { } v W ⊆  is a Bessel sequence with respect to (w.r.t.) W . If { } v f is a frame, there exists a dual frame { } * v f such that International Conference on Materials Engineering and Information Technology Applications (MEITA 2015) © 2015. The authors Published by Atlantis Press 245 W ∀Υ∈ , * * , , v v v v v v f f f f ∈Λ ∈Λ Υ = Υ = Υ ∑ ∑ (2) To state our results, the Fourier transform of an integrable function ( ) 1 2 ( ) f x L R ∈ is defined by ( ) ( ) ( ) 2 2 2 , ix R Ff f f x e dx R π ω ω ω ω ∧ − = = ∈ ∫ (3) which, as usual, can be naturally extended to functions in 2 2 ( ) L R . For a sequence ( ) { } ( ) 2 c c v Z = ∈ , we define its discrete-time Fourier transform as the function in ( ) 2 0,1 L by 2 2 ( ) ( ) ( ) ix v Z F C c v e dx c π ω ω ω − ∈ = =∑ (4) For any 2 ( ) ( ) f t L R ∈ , the novel fractional Fourier transformis defined to be ( ) { ( )}( ) ( ) ( , ) F u f t u f t u t dt α α α = = ∫  F K ( ) 2 2 0 0 [( ( ) )/2]cot /2 , , 0 0 0 ( )= m i t na b m m m n t a a t nb e α α ψ ψ − − − − − A system , { } { } m n na mb g T E g = is a Gabor frame or Weyl-Heisenberg frame for 2 2 ( ) L R , if there exist two constants , 0 A B > such that 2 2 2 2 , , , m n m n Z A g B ∈ ≤ ≤ ∑    holds for all 2 2 ( ) L R ∀ ∈  . For a Gabor frame , { } m n g the analysis mapping (also called Gabor transform) g U , given by , , : { , } g m n m n U f f g → < > , 2 2 ( ) L R f ∈ ∀ and its adjoint, the synthesis mapping (also called Gabor expansion) g U ∗ , given by , 2 2 , , , , :{ } , { } ( ) m n g m n m n m n m n U c c g c Z ∗ → ∀ ∈ ∑  are bounded linear operators. The Gabor frame operator g S is defined by g g g S U U ∗ = . Explicitly,

The Characterization for Symmetric Tight Frames and Periodic Gabor Frames and Applications in Computer Engineering