# Weighted Optimal Quadrature Formulas in Sobolev Space and Their Applications
## 2 Compound Quadrature Formulas of Hermite Type  
Let us consider quadrature formulas of the form  
\[\int_{0}^{1}p(x)\varphi(x)dx\cong\sum_{\beta=0}^{N}\sum_{\
u=0}^{t}C_{\beta}^ {(\
u)}\varphi^{(\
u)}(x_{\beta}) \tag{3}\]  
in the space \(L_{2}^{(m)}(0,1)\). Here \(L_{2}^{(m)}(0,1)\) is the space of functions whose \(m^{th}\) generalized derivative sums to square on the interval \([0,1]\); \(p(x)\) is the weight function whose  
\[\int_{0}^{1}p(x)dx<\infty;\]  
\(C_{\beta}^{(\
u)}\) are coefficients; \(x_{\beta}\) are nodes of quadrature formulas; and \(\beta=\overline{0,N},\
u=\overline{0,t},t=\overline{0,m-1}\). Here, the integral is considered to be regular, singular, fractional and strongly oscillating. The error of the quadrature formula is the difference between the integral and the sum  
\[(\ell_{N}^{(t)},\varphi)=\int_{0}^{1}p(x)\varphi(x)dx-\sum_{\beta=0}^{N}\sum_{v=0 }^{t}C_{\beta}^{(v)}\varphi^{(v)}(x_{\beta})=\int_{0}^{1}\ell_{N}^{(t)}(x)\varphi (x)dx,\]  
where  
\[\ell_{N}^{(t)}(x)=\mathcal{E}_{[0,1]}(x)p(x)-\sum_{\beta=0}^{N}\sum_{v=0}^{t}( -1)^{v}C_{\beta}^{(v)}\delta^{(v)}(x-x_{\beta}), \tag{4}\]  
\(\mathcal{E}_{[0,1]}\) is the characteristic function of the interval \([0,1]\), \(\delta(x)\) is [PERSON]'s delta function, \(\ell_{N}^{(t)}\)is the error functional of the quadrature Formula (3). The functional \(\ell_{N}^{(t)}(x)\) of the form (4) is defined in the space \(L_{2}^{(m)}(0,1)\), i.e., this functional belongs to the conjugate space \(L_{2}^{(m)*}(0,1)\), then we have  
\[(\ell_{N}^{(t)},x^{a})=0,\alpha=0,1,\ldots,m-1. \tag{5}\]  
The problem of constructing an optimal quadrature formula of the form (3) with the error functional (4) in the space \(L_{2}^{(m)}(0,1)\) consists of finding the value of  
\[\|\ell_{N}^{(t)}\|L_{2}^{(m)*}(0,1)\|^{2}=\inf_{C_{\beta}^{(v)}}|(\ell_{N}^{( t)},\psi_{\ell})| \tag{6}\]  
at fixed nodes \(x_{\beta}\). In Formula (6), \(\psi_{\ell}(x)\) is the extremal function of the quadrature Formula (3) in the space \(L_{2}^{(m)}(0,1)\). Theorem 1: _The extremal function of the error functional \(\ell_{N}^{(t)}(x)\) in the space \(L_{2}^{(m)}(0,1)\) is of the form_  
\[\psi_{\ell}(x)=(-1)^{m}\ell_{N}^{(t)}(x)*G_{m}(x)+P_{m-1}(x), \tag{7}\]  
_where \(G_{m}(x)\) is [PERSON]'s function of the operator \(\frac{d^{2m}}{dx^{2m}}\), i.e.,_  
\[G_{m}(x)=\frac{x^{2m-1}sign(x)}{2(2m-1)!},\]  
\(P_{m-1}(x)\) _is some polynomial of degree \(m-1\)._  
At \(t=0\), i.e., for the functional  
\[\ell(x)=\mathcal{E}_{[0,1]}(x)-\sum_{\beta=0}^{N}C_{\beta}\delta(x-h\beta)\]  
the extremal function was found in [1; 2]. For any \(0\leq t\leq m-1\), Theorem 1 is proved in [8]. Since \(L_{2}^{(m)}(0,1)\) is a Hilbert space, the norm of the error functional \(\ell_{N}^{(t)}\) is connected to the function \(\psi_{\ell}(x)\) through the following relation:  
\[\|\ell_{N}^{(t)}/L_{2}^{(m)*}(0,1)\|^{2}=\int_{0}^{1}(\psi_{\ell}^{(m)}(x))^{2}dx. \tag{8}\]  
In addition, there is an equality  
\[\int_{0}^{1}p(x)dx<\infty;\]0By substituting the extremal function given in Formula (7) into equality (9) and taking into account Expression (5), a series of calculations yields the squared norm of the error functional (4) corresponding to the quadrature Formula (3). \[\int_{0}^{1}p(x)dx<\infty;\]1  
\[\int_{0}^{1}p(x)dx<\infty;\]2  
\[\int_{0}^{1}p(x)dx<\infty;\]3  
where  
\[\int_{0}^{1}p(x)dx<\infty;\]4  
Recall that the coefficients \(C_{\beta}^{(v)}\) in equality (10) must satisfy the system of linear equations  
\[\int_{0}^{1}p(x)dx<\infty;\]5  
which is equivalent to the accuracy conditions (5) of the quadrature formula for polynomials of degree less than \(m\). In system (12)  
\[x_{\beta}^{(k-v)}= \begin{cases}x_{\beta}^{(k-v)},&\text{if }k-v\geq 0,\\ 0,&\text{if }k-v<0.\end{cases} \]  
Now let us formulate the conditions of the quadratic function \(F(C)\) on the set of vectors  
\[C=(C_{0}^{(0)},C_{1}^{(0)},\cdots,C_{N}^{(0)},C_{0}^{(1)},C_{1}^{(1)},\cdots, C_{N}^{(1)},\cdots,C_{0}^{(t)},C_{1}^{(t)},\cdots,C_{N}^{(t)})\]  
subject to relations (5).