# Weighted Optimal Quadrature Formulas in Sobolev Space and Their Applications
## 5 Optimal Coefficients of Weighted Quadrature Formulas
 \tag{46}\]  
By virtue of Formulas (43) and (44), we find the optimal coefficients \(\overset{\circ}{C}^{(k)}[\beta]\) at \(\beta=\overline{0,N}\):  
\[\overset{\circ}{C}^{(k)}[\beta]=hD_{1}[\beta]*u[\beta]=h\sum_{\gamma=-\infty }^{\infty}D_{1}[\beta-\gamma]u[\gamma]=\]  
\[h[(-1)^{k}\sum_{\gamma=0}^{N}D_{1}[\beta-\gamma]F_{k}[\gamma]\]  
\[+\sum_{\gamma=-\infty}^{-1}D_{1}[\beta-\gamma](-\frac{h\gamma p_{k}}{2}+(-1)^ {k}F_{k}[0])\]  
\[+\sum_{\gamma=N+1}^{\infty}D_{1}[\beta-\gamma](\frac{h\gamma p_{k}}{2}+(-1)^{ k}F_{k}[N]-\frac{p_{k}}{2})],k=\overline{0,m-1}.\]  
Hence, using Formula (45), we obtain  
\[\overset{\circ}{C}^{k}[0]=\frac{p_{k}}{2}+(-1)^{k}h^{-1}[F_{k}[1]-F_{k}[0]], \tag{47}\]  
\[\overset{\circ}{C}^{k}[\beta]=(-1)^{k}h^{-1}[F_{k}[\beta-1]-2F_{k}[\beta]+F_{ k}[beta+1]],\,\beta=\overline{0,N}, \tag{48}\]  
\[\overset{\circ}{C}^{k}[N]=\frac{p_{k}}{2}+(-1)^{k}h^{-1}[F_{k}[N-1]-F_{k}[N]],k=\overline{0,m-1} \tag{49}\]  
Thus, the following theorem is proved. Theorem 3: The optimal coefficients of the quadrature formula of the form (26) in the Sobolev space \(L^{(m)}_{2}(0,1)\) are determined by Formulas (47)-(49). From Theorem 3, when \(p(x)\equiv 1\) we obtain:  
Corollary 1: The optimal coefficients of the quadrature formula of the form (26) at \(p(x)\equiv 1\) in the Sobolev space \(L^{(m)}_{2}(0,1)\) are defined by the formulas  
\[\overset{\circ}{C}^{(0)}[0]=\frac{h}{2},\overset{\circ}{C}^{(0)}[\beta]=h, \overset{\circ}{C}^{(0)}[N]=\frac{h}{2},\]  
\[\overset{\circ}{C}^{(k)}[0]=\frac{(-1)^{k}h^{k+1}B_{k+1}}{(k+1)!},\]  
\[\overset{\circ}{C}^{(k)}[\beta]=0,\beta=\overline{1,N-1},\]  
\[h[(-1)^{k}\sum_{\gamma=0}^{N}D_{1}[\beta-\gamma]F_{k}[\gamma]\]0  
_where \(B_{k+1}\) are the Bernoulli numbers._Corollary 1 demonstrates that the optimal coefficients coincide with those of the classical Euler-Maclaurin quadrature formula. The optimality of the Euler-Maclaurin quadrature formula in the space \(L^{(m)}_{2}(0,1)\) has been established in [8; 9]. For completeness, we give the following theorem. **Theorem 4**. : _Euler-Maclorean quadrature formula_  
\[h[(-1)^{k}\sum_{\gamma=0}^{N}D_{1}[\beta-\gamma]F_{k}[\gamma]\]1  
_with the error functional_  
\[h[(-1)^{k}\sum_{\gamma=0}^{N}D_{1}[\beta-\gamma]F_{k}[\gamma]\]2  
_is the optimal quadrature formula in Sobolev space \(L^{(m)}_{2}(0,1)\). The square of the norm of the error functional of the optimal Euler-Maclorean quadrature formula is defined by the following equality_  
\[h[(-1)^{k}\sum_{\gamma=0}^{N}D_{1}[\beta-\gamma]F_{k}[\gamma]\]3  
_Here \(h=\frac{1}{N}\), \(N=2,3,\cdots,B_{a+1}\) are Bernoulli numbers, \(B_{1}=-\frac{1}{2}\),_  
\[\delta[\beta]= \begin{cases}0&\text{at }[\beta]\
eq 0,\\ 1&\text{at }[\beta]=0.\end{cases} \]  
\[\frac{B_{a+1}}{(\alpha+1)! }= \begin{cases}-(-1)^{\frac{\alpha+1}{2}}\sum_{ \gamma\
eq 0}\frac{1}{(2\pi\gamma)^{a+1}}&\text{if }\alpha+1-\text{even number},\\ 0&\text{if }\alpha+1-\text{odd number}.\end{cases} \]  
From Theorem 3, after some simplifications, we obtain the following results in the spaces \(L^{(1)}_{2}(0,1),L^{(2)}_{2}(0,1),L^{(3)}_{2}(0,1)\). **Theorem 5**. : _In the Sobolev space \(L^{(1)}_{2}(0,1)\), the following quadrature formula is the optimal quadrature formula_  
\[\int_{0}^{1}p(x)\varphi(x)dx\cong\sum_{\beta=0}^{N}\overset{\circ}{C}^{(0)}[ \beta]\varphi[\beta],\]  
_where_  
\[\overset{\circ}{C}^{(0)}[0]=h^{-1}\int_{0}^{h}p(x)(h-x)dx,\]  
\[\overset{\circ}{C}^{(0)}[\beta]=h^{-1}\int_{h(\beta-1)}^{h(\beta+1)}p(x)[(h \beta-x)sign(x-h\beta)+h]dx,\beta=\overline{1,N-1},\]  
\[\overset{\circ}{C}^{(0)}[N]=h^{-1}\int_{1-h}^{1}p(x)(x-1+h)dx.\]  
**Theorem 6**.