# Weighted Optimal Quadrature Formulas in Sobolev Space and Their Applications
## 5 Optimal Coefficients of Weighted Quadrature Formulas  
with Derivative  
We now solve the system of linear algebraic Equations (34) and (35) with respect to \(\overset{\circ}{C}_{\beta}^{(k)}\) (\(\beta=\overline{0,N},k=\overline{0,m-1}\)), the optimal coefficients of weight quadrature formulas with derivatives. Let us rewrite systems (34) and (35) in the following form  
\[\sum_{\gamma=0}^{N}\overset{\circ}{C}_{\gamma}^{(k)}G_{1}[\beta](h\beta-h \gamma)+(-1)^{k}k!\lambda_{k}=(-1)^{k}F_{k\beta}, \tag{37}\]  
\[\sum_{\gamma=0}^{N}\overset{\circ}{C}_{\gamma}^{(k)}=p_{k},\beta=\overline{0,N},k=\overline{0,m-1}. \tag{38}\]  
Here \(\lambda_{k}\) is a constant, and \(F_{k\beta}\) is determined by the equality (36),  
\[p_{k}=\frac{g_{k}}{k!}-\sum_{i=0}^{k-1}\sum_{\gamma=0}^{N}\overset{\circ}{C}  
The coefficients of the first term of the first equation depend only on the difference \((h\beta-h\gamma)\). These kinds of equations in the continuous case, where instead of the sum there are integrals, are called Wiener-Hopf equations. As is typical for Wiener-Hopf equations, we assume that \(\overset{\circ}{C}^{(k)}_{\gamma}\) is defined everywhere, i.e., \(\gamma\in Z\), and equal to zero if \(h\gamma\
ot\in[0,1]\).  
Let us further assume \(\overset{\circ}{C}^{(k)}_{\beta}=\overset{\circ}{C}^{(k)}[\beta],G_{1}[\beta] =G_{1}(h\beta),F_{k\beta}=F_{k}[\beta]\). Then systems (37) and (38) are written in the form of convolution equations  
\[G_{1}[\beta]*\overset{\circ}{C}^{(k)}[\beta]+(-1)^{k}k!\lambda_{k}=(-1)^{k}F_ {k}[\beta],\beta=\overline{0,N}, \tag{39}\]  
\[\sum_{\gamma=0}^{N}\overset{\circ}{C}^{(k)}[\gamma]=p_{k}, \tag{40}\]  
\[\overset{\circ}{C}^{(k)}[\beta]=0,[\beta]\
ot\in[0,1]. \tag{41}\]  
We now proceed to the actual solution of systems (39) and (41). For this purpose, instead of \(\overset{\circ}{C}^{(k)}[\beta]\) we introduce the following function  
\[u[\beta]=G_{1}[\beta]*\overset{\circ}{C}^{(k)}[\beta]+(-1)^{k}k!\lambda_{k}.\]  
Next, we define the function \(u[\beta]\) when \(\beta\leq 0\) and \(\beta\geq N\). Let \(\beta\leq 0\), then by virtue of (40) we obtain  
\[u[\beta]=-\frac{h\beta p_{k}}{2}+a_{k}^{-},\]  
\(a_{k}^{-}\) is unknown. Let \(\beta\geq N\), then  
\[u[\beta]=\frac{h\beta p_{k}}{2}+a_{k}^{+},\]  
\(a_{k}^{+}\) is unknown. So, we have defined a function \(u[\beta]\) for all values of \([\beta]\in Z\):  
\[u[\beta]= \begin{cases}-\frac{h\beta p_{k}}{2}+a_{k}^{-}&\text{ if }\beta\leq 0, \\ (-1)^{k}F_{k}[\beta]&\text{ if }0\leq\beta\leq N,\\ \frac{h\beta p_{k}}{2}+a_{k}^{+}&\text{ if }\beta\geq N.\end{cases}  \tag{42}\]  
Since in (42) at \(\beta=0\) and \(\beta=N\) the left and right parts coincide, we have  
\[a_{k}^{-}=(-1)^{k}F_{k}[0],\;a_{k}^{+}=(-1)^{k}F_{k}[N]-\frac{p_{k}}{2}.\]  
Then we obtain a new representation of the function \(u[\beta]\) in \([\beta]\in Z\):  
\[u[\beta]= \begin{cases}-\frac{h\beta p_{k}}{2}+(-1)^{k}F_{k}[0]&\text{ if } \beta\leq 0,\\ (-1)^{k}F_{k}[\beta]&\text{ if }0\leq\beta\leq N,\\ \frac{h\beta p_{k}}{2}+(-1)^{k}F_{k}[N]-\frac{p_{k}}{2}&\text{ if }\beta\geq N.\end{cases}  \tag{43}\]  
Now we will need the well-known formula [9]:  
\[hD_{1}[\beta]*G_{1}[\beta]=\delta[\beta], \tag{44}\]where  
\[D_{1}[\beta]= \begin{cases}0&\text{ if }[\beta]\geq 2,\\ h^{-2}&\text{ if }[\beta]=1,\\ -2h^{-2}&\text{ if }[\beta]=0,\end{cases}  \tag{45}\]  
\[\delta[\beta]= \begin{cases}0&\text{ at }[\beta]\
eq 0,\\ 1&\text{ at }[\beta]=0.\end{cases}