## 3 The Proposed S-LCMV Cost Function  
This section analyzes the sparsity of \(\mathbf{g}(t,\)\(d_{m})\) in detail and reconstructs the corresponding cost function with a sparse \(\mathbf{g}(t,\)\(d_{m})\) to provide sufficient DoFs. Based on the aforementioned definition of the vector \(\mathbf{g}(t,\)\(d_{m})\), we fixed \(s\) and \(q\) in \(g_{s,q,p}(t,\)\(d_{m})\) to analyze the variation in the values of \(g_{s,q,p}(t,\)\(d_{m})\) with respect to the Legendre polynomial stage \(p\). Referring to the expression of \(g_{s,q,p}(t,\)\(d_{m})\), we now examine the \(p\)-dependent variations of \(A_{p}x_{1}C_{1}I_{0}(x_{1}r)\). According to the boundary continuity condition, the important factor \(C_{1}\) can be obtained as follows [30]:  
\[C_{1}=\mathbf{Y}_{1}(1,2)+\mathbf{Y}_{2}(1,1)+\mathbf{Y}_{3}(1,1) \tag{6}\]  
\[\mathbf{Y}_{1}=\mathbf{A}_{1}{}^{-1}\times\mathbf{B}_{1}\times\mathbf{A}_{2}{ }^{-1}\times\mathbf{B}_{2}\times\mathbf{A}_{3}{}^{-1}\times\mathbf{B}_{3} \times\mathbf{A}_{4}{}^{-1}\times\mathbf{B}_{4}\times\mathbf{A}_{5}{}^{-1} \times\mathbf{B}_{5}\times\mathbf{A}_{6}{}^{-1}\times\mathbf{B}_{6} \tag{7}\]  
\[\mathbf{Y}_{2}=\mathbf{A}_{1}{}^{-1}\times\mathbf{B}_{1}\times\mathbf{A}_{2}{ }^{-1}\times\mathbf{O}_{2} \tag{8}\]  
\[\mathbf{Y}_{3}=\mathbf{A}_{1}{}^{-1}\times\mathbf{O}_{1} \tag{9}\]  
where \(\mathbf{Y}(i,j)\) denotes the element of matrix \(\mathbf{Y}\) located in the \(i\)-th row and \(j\)-th column. The matrix \(\mathbf{A}_{j}{}^{-1}\) represents the inverse of matrix \(\mathbf{A}_{j}\) with  
\[\mathbf{A}_{j}=[ \begin{matrix}\mu_{j}I_{1}(x_{j}r_{j})&\mu_{j}K_{1}(x_{j}r_{ j})\\ \times I_{0}(x_{j}r_{j})&-x_{j}K_{0}(x_{j}r_{j})\end{matrix} ] \tag{10}\]\[\mathbf{B}_{j}=[ \begin{matrix}\mu_{j+1}I_{1}(x_{j+1}r_{j})&\mu_{j+1}K_{1}(x_{j+1}r _{j})\\ x_{j+1}I_{0}(x_{j+1}r_{j})&-x_{j+1}K_{0}(x_{j+1}r_{j})\end{matrix} ] \tag{11}\]  
\[\mathbf{O}_{1}=[ \begin{matrix}\mu_{2}K_{1}(x_{2}r_{0})I_{1}(x_{2}r_{0})\\ x_{2}K_{1}(x_{2}r_{0})I_{0}(x_{2}r_{0})\end{matrix} ] \tag{12}\]  
\[\mathbf{O}_{2}=[ \begin{matrix}-\mu_{2}K_{1}(x_{2}r_{0})I_{1}(x_{2}r_{0})\\ x_{2}K_{0}(x_{2}r_{0})I_{1}(x_{2}r_{0})\end{matrix} ] \tag{13}\]  
In the above equations, \(I_{0}(xr)=\mathit{besseli}(0,\,\,\mathit{x}r)\) and \(I_{1}(xr)=\mathit{besseli}(1,\,\,\mathit{x}r)\) are modified Bessel functions of the first kind with order \(0\) and order\(1\), respectively. \(K_{0}(xr)=\mathit{besselk}(0,\,\,\mathit{x}r)\) and \(K_{1}(xr)=\mathit{besselk}(1,\,\,\mathit{x}r)\) are modified Bessel functions of the second kind with order \(0\) and order\(1\), respectively. When \(p\) becomes sufficiently large, \(\lambda>>k_{j}\), then we have \(x_{j}=(\lambda^{2}-k_{j}^{2})^{1/2}\approx\lambda\), \(I_{1}(x_{j}r_{j})\approx I_{1}(x_{j+1}r_{j})\), \(K_{1}(x_{j}r_{j})\approx K_{0}(x_{j}r_{j})\), \(I_{0}(x_{j}r_{j})\approx I_{0}(x_{j+1}r_{j})\), and \(K_{0}(x_{j}r_{j})\approx K_{0}(x_{j+1}r_{j})\). For further analysis, we show the distributions of the Bessel functions in Figure 2. The Bessel functions of the first and second kinds exponentially grow and decay, respectively, with increasing parameter \(\mathit{x}r\). Therefore, when \(\lambda\) is large (i.e., the stage of the Gauss-Legendre integral is high or \(p\) is large), the elements in the second row of \(\mathbf{A}_{j}-^{1}\) and the elements in the first column of \(\mathbf{B}_{j}\) are dominant, while the other elements are negligibly small (approximately zero). To facilitate the analysis, the variables \(\alpha\), \(\beta\), \(a\), and \(b\) are introduced to further simplify the expression. Assuming that \(\alpha\) and \(\beta\) are infinitesimally small (approaching zero) and much smaller than \(a\) and \(b\), the elements in \(\mathbf{A}_{j}-^{1}\) and \(\mathbf{B}_{j}\) can be expressed as  
\[\mathbf{A}_{j}^{-1}= \begin{bmatrix}a_{j}^{\prime}&a_{j}^{\prime\prime}\\ a_{j}^{\prime}&a_{j}^{\prime\prime}\end{bmatrix}  \tag{14}\]  
\[\mathbf{B}_{j}= \begin{bmatrix}b_{j}^{\prime}&\beta_{j}^{\prime}\\ b_{j}^{\prime}&\beta_{j}^{\prime\prime}\end{bmatrix}