# Legendre Multiwavelet Transform and Its Application in Bearing Fault Detection
## 2 Legendre Multiwavelet Transform  
### Legendre Multiwavelet  
Legendre multiwavelet is constructed by Legendre polynomials with can described as  
\[L_{0}(x) =1, \tag{1}\] \[L_{1}(x) =x,\] \[L_{k+2}(x) =\frac{2k+3}{k+2}xL_{k+1}(x)-\frac{k+1}{k+2}L_{k}(x)\]  
where \(k=0,1,\cdots,p-1\) and \(p\) represent the number of adopted Legendre polynomials, respectively. The Legendre scale basis function \(\phi_{k}(x)\) is described by [31] as the following:  
\[\phi_{k}(x)=\begin{cases}\sqrt{2k+1}L_{k}(2x-1),&x\in[0,1]\\ 0,&x\
otin[0,1]\end{cases} \tag{2}\]  
In general, a subspace \(V_{p,n}\) comprising piecewise polynomials is defined as  
\[V_{p,n}= \{f:f|_{I_{nl}}\text{is a polynomial of degree} \tag{3}\] \[\text{strictly less than }p;f\text{ vanishes elsewhere}\}\]  
the arrangement of which constitutes a linear space. Here, the variables \(n=0,1,\ldots,N\) denote the resolution level, while \(l=0,1,\ldots,2^{n}-1\) represents the translation parameter. The corresponding interval \(I_{nl}\) is defined as \(I_{nl}=[2^{-n}l,2^{-n}(l+1)]\). It is apparent that the set \(\phi_{k}{}^{p-1}k=0\) establishes an orthonormal basis for the subspace \(Vp,0\). Subsequently, the construction of the subspace \(V_{p,n}\) is also achieved utilizing \(\phi_{k}\) through dilation and translation.  
\[V_{p,n}=span\Big{\{}\phi_{k,nl}(x)=2^{\frac{n}{2}}\phi_{k,n}(2^{n}x-l)\Big{\}} \tag{4}\]  
which forms an orthonormal basis in the subspace \(V_{p,n}\).  
If the analysis of vibration signals related to various bearing faults is confined exclusively within the subspace \(V_{p,n}\), it efficiently captures the low-frequency components at resolution level \(n\). However, numerous characteristic features of bearing faults prominently manifest within the high-frequency components. Therefore, it becomes imperative to define the orthogonal complement of the subspace \(V_{p,n}\) within \(V_{p,n+1}\), termed as the multiwavelet subspace \(W_{p,n}\). This necessity arises from the equation as  
\[V_{p,n}\oplus W_{p,n}=V_{p,n+1},\qquad V_{p,n}\perp W_{p,n} \tag{5}\]  
[PERSON] et al. [31] developed the multiwavelet subspace to facilitate the efficient computation of integral and differential operators. The implications of this construction can be elucidated through the two scale relations expressed as  
\[\phi_{k}(x) =\sqrt{2}\sum_{k^{\prime}=0}^{p-1}(h_{kk}^{(0)}\phi_{k^{\prime}}(2 x)+h_{kk^{\prime}}^{(1)}\phi_{k^{\prime}}(2x-1)) \tag{6}\] \[\psi_{k}(x) =\sqrt{2}\sum_{k^{\prime}=0}^{p-1}(g_{kk^{\prime}}^{(0)}\phi_{k^{ \prime}}(2x)+g_{kk^{\prime}}^{(1)}\phi_{k^{\prime}}(2x-1)) \tag{7}\]where \(\psi_{k}\) is the multiwavelet basis. In this paper, the coefficient matrices mentioned above, \(H=(h_{ij}^{(0)},h_{ij}^{(1)})_{p\times 2p}\) and \(G=(g_{ij}^{(0)},g_{ij}^{(1)})_{p\times 2p}\), are utilized for convolving the raw bearing data. This convolution process is convenient and facilitates the comprehensive extraction of various characteristics associated with faults present in the bearing.  
Specifically, diverse regularities are better suited for adaptively discerning complex fault characteristics and offering alternative methods for extracting features. These methods require less expertise compared to traditional wavelet transform techniques.