Splitting of Framelets and Framelet Packets

Due to resilience to background noise, stability of sparse reconstruction, and ability to capture local time-frequency information, the frame theory is becoming a dynamic forefront topic in data science. In this study, we overcome the disadvantages in the construction of traditional framelet packets derived by frame multiresolution analysis and square iterative matrices. We propose two novel approaches: One is to directly split known framelets again and again; the other approach is based on a generalized scaling function whose shifts are not a frame of some space. In these two approaches, the iterative matrices used are not square and the number of rows in the iterative matrix can be any integer number.


Introduction
The notion of wavelet packets was first introduced by Coifman, Meyer, and Wickerhauser [1]. A wavelet packet is a library from which various orthonormal bases can be picked. Wavelet packets can extract better time-frequency features than wavelets, so wavelet packets play an important role in the application of wavelets. Later, for the highdimensional case, the tensor product wavelet packets were constructed by Chui and Li [2], whereas the non-tensor product wavelet packets were constructed by Shen [3].
As a generalization of wavelet packets, framelet packets was first constructed from frame multiresolution analyses (FMRA) [4][5][6]. Since generally the scaling function of FMRA is discontinuous in frequency domain, the derived framelet packets cannot possess nice time-domain localization. Moreover, the iterative matrix in the construction of framelet packets is square, and only when the matrix is unitary, the iterative process can be operated up to infinitely many times which can lead to framelet packet with finer and finer frequency bands.
In order to solve the above problems, we propose two approaches. One approach is to abandon multiresolution structure used in traditional construction of framelet packet. Instead, we will directly split framelets by various iterative matrices. The other approach is to remove the use of scaling function in FMRA, i.e., it starts from a generalized scaling function whose shifts are not a frame of some space. In these two approaches, all the iterative matrices are not square and the number of rows in iterative matrix can be any integer number. The framelet packets constructed by us possess fine properties, including short supported, high approximation orders, symmetry, and smoothness.

Preliminaries
Denote the space of square-integrable functions on R d by L 2 (R d ) and the space of 2πZ d −periodic bounded function by L ∞ (T d ). Denote the inner product by (·, ·) and the norm by · . We define the Fourier transform f of f ∈ L 1 (R d ) by We denote the set of vertexes of the cube [0, 1] d by {0, 1} d . The notation δ ij is the Kronecker delta symbol, i.e., δ ij = 0 (i = j) and δ ii = 1. Let {h n } ∞ 1 be a sequence in L 2 (R d ). If there exists a B > 0 such that If the affine system {ψ µ,j,k } is a frame for L 2 (R d ), then the set {ψ µ } l 1 is called a framelet.
Generally, when m is large, the obtained frame in Theorem 2 has finer time-frequency localization, but at the same time, the frame bounds become very large. Therefore, only when λ = Λ = 1, the above splitting trick can be operated for infinite many times, i.e., m can trend to infinity and the bounds of the obtained frames are still A and B .

Framelet packets
Generally, since the scaling function of FMRA is discontinuous in the frequency domain, the derived framelet packets cannot possess nice time-domain localization [4][5][6]. In order to solve the above problems, we remove the restriction of FMRA and square iterative matrix, i.e., we start from a generalized scaling function whose shifts are not a frame of some space and the number of rows in iterative matrix can be any integer number.

Conclusions
In this study, the role of frame multiresolution analysis and square iterative matrices in the construction of frame packets is removed. Two novel tricks are proposed to construct framelets with better time-frequency localization features than those of known framelets. One approach is to split known framelets by using various non-square iterative matrices. The other approach is to start from a generalized scaling function. Moreover, the iterative process in these two approaches can be operated an infinite amount of times.