## 4 Approximation Properties  
In this section, we study the approximation properties of the integro quintic spline \(s\) obtained in Section 3. For \(k=0,1,\ldots,5\), we use \(y^{(k)}_{j}\) to denote \(y^{(k)}(x_{j})\), \(j=0,1,\ldots,n\). For \(k=0,1,2,3,4\), we use \(s_{j}\), \(m_{j}\), \(M_{j}\), \(T_{j}\) and \(F_{j}\) to denote \(s^{(k)}(x_{j})\), \(j=0,1,\ldots,n\). In addition, we define:  
\[W_{j}:=\frac{F_{j+1}-F_{j-1}}{2h}\]  
in order to approximate \(y^{(5)}(x_{j})\), \(j=1,2,\ldots,n-1\). For \(j=0,1,\ldots,n\),  
\[s_{j} = \frac{1}{120}(c_{j-2}+26c_{j-1}+66c_{j}+26c_{j+1}+c_{j+2}),\] \[m_{j} = \frac{1}{24h}(-c_{j-2}-10c_{j-1}+10c_{j+1}+c_{j+2}),\] \[M_{j} = \frac{1}{6h^{2}}(c_{j-2}+2c_{j-1}-6c_{j}+2c_{j+1}+c_{j+2}),\] \[T_{j} = \frac{1}{2h^{2}}(-c_{j-2}+2c_{j-1}-2c_{j+1}+c_{j+2}),\] \[F_{j} = \frac{1}{h^{4}}(c_{j-2}-4c_{j-1}+6c_{j}-4c_{j+1}+c_{j+2}).\]Moreover,  
\[W_{j} = \frac{1}{2h^{5}}(-c_{j-3}+4c_{j-2}-5c_{j-1}+5c_{j+1}-4c_{j+2}+c_{j+3} ),\;\;j=1,2,\ldots,n-1.\]  
By using (24), (25) and the above results, we can get some important relations between \(s_{j}\), \(m_{j}\), \(M_{j}\), \(T_{j}\), \(F_{j}\), \(W_{j}\) and \(I_{j}\) of the integro quintic spline. We list the relations as follows. (Set I)  
\[347m_{0}+1044m_{1}+225m_{2}+4m_{3}=\frac{1}{h^{2}}(120I_{2}+1110I_{1}-690I_{0})- \frac{540}{h}s_{0}-54 hM_{0}; \tag{28}\]  
for \(j=2,3,\ldots,n-2\),  
\[m_{j-2}+56m_{j-1}+246m_{j}+56m_{j+1}+m_{j+2}=\frac{30}{h^{2}}(-I_{j-2}-9I_{j-1} +9I_{j}+I_{j+1}); \tag{29}\]  
\[4m_{n-3}+225m_{n-2}+1044m_{n-1}+347m_{n}=\frac{1}{h^{2}}(690I_{n-1}-1110I_{n-2 }-120I_{n-3})+\frac{540}{h}\theta_{n}. \tag{30}\]  
(Set II)  
For \(j=0,1,\ldots,n-3\),  
\[s_{j} = \frac{1}{120}(c_{j-2}+26c_{j-1}+66c_{j}+26c_{j+1}+c_{j+2}),\]0  
for \(j=3,4,\ldots,n\),  
\[s_{j} = \frac{1}{120}(c_{j-2}+26c_{j-1}+66c_{j}+26c_{j+1}+c_{j+2}),\]1  
(Set III)  
\[s_{j} = \frac{1}{120}(c_{j-2}+26c_{j-1}+66c_{j}+26c_{j+1}+c_{j+2}),\]2  
\[s_{j} = \frac{1}{120}(c_{j-2}+26c_{j-1}+66c_{j}+26c_{j+1}+c_{j+2}),\]3  
for \(j=3,4,\ldots,n\),  
\[ \begin{array}{rcl}s_{j}&=&-s_{j-3}+\frac{1}{18h}(47I_{j-3}-58I_{j-2}+47I_{j- 1})\\ &&+\frac{h}{540}(-61m_{j-3}+423m_{j-2}-423m_{j-1}+61m_{j}).\end{array}  \tag{35}\]  
(Set IV)  
\[M_{0}=\tilde{y}_{0}^{{}^{\prime\prime}}=\frac{10}{h^{3}}I_{0}-\frac{10}{h^{2}} s_{0}-\frac{13}{3h}m_{0}-\frac{2}{3h}m_{1}-\frac{h}{9}T_{0}+\frac{h}{36}T_{1}; \tag{36}\]  
for \(j=1,2,\ldots,n\),  
\[M_{j} = -\frac{10}{h^{3}}I_{j-1}+\frac{10}{h^{2}}s_{j-1}+\frac{7}{3h}m_{j -1}+\frac{8}{3h}m_{j}-\frac{h}{18}T_{j-1}+\frac{5h}{36}T_{j}. \tag{37}\]  
(Set V)  
\[F_{0}=-\frac{120}{h^{5}}I_{0}+\frac{120}{h^{4}}s_{0}+\frac{40}{h^{3}}m_{0}+ \frac{20}{h^{3}}m_{1}-\frac{11}{3h}T_{0}-\frac{4}{3h}T_{1}; \tag{38}\]  
for \(j=1,2,\ldots,n\),  
\[F_{j} = \frac{120}{h^{5}}I_{j-1}-\frac{120}{h^{4}}s_{j-1}-\frac{40}{h^{3} }m_{j-1}-\frac{20}{h^{3}}m_{j}+\frac{5}{3h}T_{j-1}+\frac{10}{3h}T_{j}. \tag{39}\]  
(Set VI)  
\[W_{1}=\frac{60}{h^{6}}(I_{0}+I_{1})-\frac{60}{h^{5}}(s_{0}+s_{1})-\frac{10}{h^{4 }}(2m_{0}+3m_{1}+m_{2})+\frac{1}{6h^{2}}(11T_{0}+9T_{1}+10T_{2}); \tag{40}\]for \(j=2,3,\ldots,n-1\),  
\[W_{j} = \frac{60}{h^{6}}(I_{j}-I_{j-2})-\frac{60}{h^{5}}(s_{j}-s_{j-2})- \frac{20}{h^{4}}(m_{j}-m_{j-2}) \tag{41}\] \[-\frac{10}{h^{4}}(m_{j+1}-m_{j-1})+\frac{5}{6h^{2}}(-T_{j-2}-2T_{j -1}+T_{j}+2T_{j+1}).\]  
**Theorem 2**. : _Let \(s\) be the integro quintic spline determined by (17)-(19) with the artificial end conditions given in Section 2.