# Weighted Optimal Quadrature Formulas in Sobolev Space and Their Applications
## 2 Compound Quadrature Formulas of Hermite Type
For this purpose, we apply the method of Lagrange indefinite multipliers. Let us compose the Lagrange function  
\[F(C,\lambda)=F(C)+2\sum_{k=0}^{m-1}\lambda_{k}(\ell_{N}^{(t)},x^{k}).\]  
Equating to zero the partial derivatives of \(F(C,\lambda)\) by \(C_{\beta}^{(v)}\) and \(\lambda_{k}\), we obtain  
\[\sum_{\beta=0}^{N}\sum_{v=0}^{t}C_{\beta}^{(v)}G_{m}^{(v+v^{\prime})}(x_{\beta ^{\prime}}-x_{\beta})+\sum_{k=0}^{m-1}\lambda_{k}x_{\beta^{\prime}}^{(k-v^{ \prime})}\times\frac{k!}{(k-v^{\prime})!}=f_{\beta^{\prime}}^{(v^{\prime})}, \tag{13}\]  
\[\
u^{{}^{\prime}}=\overline{0,t},\ \beta^{\prime}=\overline{0,N},\]  
\[\sum_{\beta=0}^{N}\sum_{v=0}^{N}\frac{k!}{(k-v)!}C_{\beta}^{k-v}=g_{k},\ k= \overline{0,m-1}, \tag{14}\]  
where  
\[f_{\beta^{\prime}}^{(v^{\prime})}=\int_{0}^{1}p(x)G_{m}^{(v^{\prime})}(x-x_{ \beta^{\prime}})dx,\ \
u^{{}^{\prime}}=\overline{0,t},\ \beta^{{}^{\prime}}=\overline{0,N},\]  
\[g_{k}=\int_{0}^{1}p(x)x^{k}dx,\ k=\overline{0,m-1}.\]In the system of Equations (13) and (14), the unknown variables are \(C_{\beta}^{(\
u)}\) and \(\lambda_{k}\). The solution to this system corresponds to a stationary point of the function \(F(C,\lambda)\), which we denote by \(\overset{\circ}{C}_{\beta}^{(\
u)}\) and \(\overset{\circ}{\lambda}_{k}\). According to the theory of constrained extrema, there exists a sufficient condition ensuring that this solution represents a conditional minimum of \(F(C)\) on the manifold defined by (5). This condition requires the associated quadratic form to be positive definite. \[\phi(C)=\sum_{\beta=0}^{N}\sum_{\beta^{\prime}=0}^{N}\sum_{\
u=0}^{t}\sum_{\
u =0}^{t}\frac{\partial^{2}\ F(C,\lambda)}{\partial\overset{\circ}{C}_{\beta}^{ (\
u)}\partial C_{\beta^{\prime}}^{(\
u^{\prime})}}C_{\beta}^{(\
u)}, \tag{15}\]  
on the set of vectors \(C_{\beta}^{(\
u)}\) obtaining the requirement  
\[\sum_{\beta=0}^{N}\sum_{\
u=0}^{t}C_{\beta}^{(\
u)}x_{\beta}^{(k-\
u)}\frac{k! }{(k-\
u)!}=0,\ k=\overline{0,m-1}. \tag{16}\]  
In matrix form system (16) has the form  
\[SC=0. \tag{17}\]  
We proceed to prove that in the considered case, the quadratic form (15) is positive definite. Lemma 1: _For any nonzero vector \(C\) belonging to the subspace defined by \(SC=0\), the function \(\Phi(C)\) takes strictly positive values.\({}^{*}\)_  
Proof of Lemma 1: From the definition of the Lagrange function \(F(C,\lambda)\), and by utilizing equality (15), it directly follows that...  
\[F(C,\lambda)=F(C)+2\sum_{k=0}^{m-1}\lambda_{k}(\ell_{N}^{(t)},x^{k}).\]0  
Consider the following functional  
\[F(C,\lambda)=F(C)+2\sum_{k=0}^{m-1}\lambda_{k}(\ell_{N}^{(t)},x^{k}).\]1  
It is known that, by condition (6), this functional belongs to \(L_{2}^{(m)*}\), i.e., \(\mu(C)\in L_{2}^{(m)*}(0,1)\). For this functional, there corresponds an extremal function \(\psi_{\mu}(x)\in L_{2}^{(m)}(0,1)\), which is a solution of Equation  
\[F(C,\lambda)=F(C)+2\sum_{k=0}^{m-1}\lambda_{k}(\ell_{N}^{(t)},x^{k}).\]2  
The solution of Equation (20) has the form  
\[F(C,\lambda)=F(C)+2\sum_{k=0}^{m-1}\lambda_{k}(\ell_{N}^{(t)},x^{k}).\]3  
The square of the norm of the function \(\psi_{\mu}(x)\) in \(L_{2}^{(m)}(0,1)\) coincides with the form \(\phi(C)\)  
\[F(C,\lambda)=F(C)+2\sum_{k=0}^{m-1}\lambda_{k}(\ell_{N}^{(t)},x^{k}).\]4  
The system of Equation (24) means that \(U_{\overline{C}}^{(v)}(x)\) takes zero values at all nodes \(x_{\beta}\), i.e., \(U_{\overline{C}}^{(v)}(x_{\beta})=0\), when \(\beta=\overline{0,N},\
u=\overline{0,t}\). Then with respect to the norm in \(L_{2}^{(m)*}(0,1)\) of the functional \(\mu_{\overline{C}}(x)\), we have  
\[F(C,\lambda)=F(C)+2\sum_{k=0}^{m-1}\lambda_{k}(\ell_{N}^{(t)},x^{k}).\]5  
which is possible only at \(\overline{C}=0\). Taking this into account from (24), we obtain  
\[F(C,\lambda)=F(C)+2\sum_{k=0}^{m-1}\lambda_{k}(\ell_{N}^{(t)},x^{k}).\]6  
By convention, the matrix \(S\) has a right inverse, then \(S^{*}\) has a left inverse. Hence and from (25) it follows that \(\overline{\lambda}=0\). Lemma 2 is proved completely. Thus, systems (13) and (14) have a single solution.