# On an Optimal Quadrature Formula in a Hilbert Space of Periodic Functions
## 6 The Norm for the Error Functional of the Optimal Quadrature Formula  
Here, we present the proof of Theorem 3. For this, we simplify the expression (36) as follows:  
\[\|\hat{\ell}|\widehat{H}_{2}^{(m)^{*}}\|^{2} = \frac{2\pi}{\omega^{2m}}-\frac{1}{\omega^{2m}}\sum\limits_{k^{ \prime}=1}^{N}\hat{\mathbb{C}}_{k^{\prime}}\exp\biggl{(}\frac{2\pi\mathrm{i}k^{ \prime}\omega}{N}\biggr{)}-\frac{1}{\omega^{2m}}\sum\limits_{k=1}^{N}\hat{ \mathbb{C}}_{k}\exp\biggl{(}\frac{-2\pi\mathrm{i}k\omega}{N}\biggr{)}\] \[+\sum\limits_{k=1}^{N}\sum\limits_{k^{\prime}=1}^{N}\hat{ \mathbb{C}}_{k}\hat{\mathbb{C}}_{k^{\prime}}\sum\limits_{\beta\
eq 0}\frac{\exp \Bigl{(}\frac{2\pi\mathrm{i}\beta(k-k^{\prime})}{N}\Bigr{)}}{2\pi\beta^{2m}}= \frac{2\pi}{\omega^{2m}}-\sum\limits_{k^{\prime}=1}^{N}\hat{\mathbb{C}}_{k^{ \prime}}\] \[\times\Biggl{[}\frac{1}{\omega^{2m}}\exp\biggl{(}\frac{2\pi \mathrm{i}k^{\prime}\omega}{N}\biggr{)}-\sum\limits_{k=1}^{N}\hat{\mathbb{C}}_ {k}\sum\limits_{\beta\
eq 0}\frac{\exp\Bigl{(}\frac{2\pi\mathrm{i}(k-k^{ \prime})}{N}\Bigr{)}}{2\pi\beta^{2m}}\Biggr{]}\] \[-\frac{1}{\omega^{2m}}\sum\limits_{k=1}^{N}\hat{\mathbb{C}}_{k} \exp\biggl{(}-\frac{2\pi\mathrm{i}\omega k}{N}\biggr{)}.\]  
Hence, taking (38) into account for the square of the norm, we obtain  
\[\|\hat{\ell}|\widehat{H}_{2}^{(m)^{*}}\|^{2}=\frac{2\pi}{\omega^{2m}}-\frac{1 }{\omega^{2m}}\sum\limits_{k=1}^{N}\hat{\mathbb{C}}_{k}\exp\biggl{(}-\frac{2 \pi\mathrm{i}\omega k}{N}\biggr{)}.\]  
Then, using the analytic expressions of the optimal coefficients, i.e., by the formula (52), we obtain  
\[\|\hat{\ell}|\widehat{H}_{2}^{(m)^{*}}\|^{2} = \frac{2\pi}{\omega^{2m}}-\frac{2\pi}{N\omega^{2m}}\sum\limits_{k= 1}^{N}\biggl{(}\frac{\sin\frac{\pi\omega}{N}}{\frac{\pi\omega}{N}}\biggr{)}^{2m}\] \[\times \frac{(2m-1)!\exp\Bigl{(}\frac{2\pi\mathrm{i}k\omega}{N}\Bigr{)} \exp\Bigl{(}-\frac{2\pi\mathrm{i}k\omega}{N}\Bigr{)}}{2\sum\limits_{n=0}^{m-2} a_{n}^{(2m-2)}\cos 2\pi(m-1-n)\frac{\omega}{N}+a_{m-1}^{(2m-2)}}.\]  
Provided that \(\exp\Bigl{(}\frac{2\pi\mathrm{i}k\omega}{N}\Bigr{)}\exp\Bigl{(}-\frac{2\pi \mathrm{i}k\omega}{N}\Bigr{)}=1\) and \(\sum\limits_{k=1}^{N}1=N\), for the square of the norm, we finally obtain  
\[\|\hat{\ell}|\widehat{H}_{2}^{(m)^{*}}\|^{2} = \frac{2\pi}{\omega^{2m}}\Biggl{[}1-\biggl{(}\frac{\sin\frac{\pi \omega}{N}}{\frac{\pi\omega}{N}}\biggr{)}^{2m}\frac{(2m-1)!}{2\sum\limits_{n=0} ^{m-2}a_{n}^{(2m-2)}\cos 2\pi(m-1-n)\frac{\omega}{N}+a_{m-1}^{(2m-2)}} \Biggr{]}.\]  
Theorem 3 is completely proved.  
Now, we give the square of the norm of the error functional for the several first values of \(m\) for \(\omega<N\).  
If \(m=1\), then  
\[\left\|\hat{\ell}|\widehat{H}_{2}^{(1)*}\right\|^{2} = \frac{\pi}{3}\biggl{(}\frac{2\pi}{N}\biggr{)}^{2}-\frac{\omega^{2} }{180}\biggl{(}\frac{2\pi}{N}\biggr{)}^{4}+O\biggl{(}\biggl{(}\frac{2\pi}{N} \biggr{)}^{6}\biggr{)}.\]  
Now let \(m=2\). It follows that  
\[\left\|\hat{\ell}|\widehat{H}_{2}^{(2)*}\right\|^{2} = \frac{\pi}{\cos\frac{2\pi\omega}{N}+2}\Biggl{[}\frac{1}{5!} \biggl{(}\frac{2\pi}{N}\biggr{)}^{4}+\frac{3\omega^{2}}{7!}\biggl{(}\frac{2 \pi}{N}\biggr{)}^{6}\Biggr{]}+O\biggl{(}\biggl{(}\frac{2\pi}{N}\biggr{)}^{8} \biggr{)}.\]It is easy to see that, for \(N\to\infty\) and \(\omega<N\),  
\[\left(\frac{\sin\frac{\pi\omega}{N}}{\frac{\pi\omega}{N}}\right)^{2m} \to 1\text{ and }\] \[2\sum_{n=0}^{m-2}a_{n}^{(2m-2)}\cos\left(2\pi(m-1-n)\frac{\omega}{N} \right)+a_{m-1}^{(2m-2)}\to 2\sum_{n=0}^{m-2}a_{n}^{(2m-2)}+a_{m}^{(2m-2)}=(2m-1)!,\] \[\text{ then }\|\ell\|^{2}\to 0.\]  
In the case \(\omega>N\) and \(\omega\to\infty\), we have  
\[\|\ell\|^{2}\to\frac{2\pi}{\omega^{2m}}.\]  
From the last results, we can conclude that, for the functions from the space \(\hat{H}_{2}^{m}\), the rate of convergence of the constructed optimal quadrature formula is \(O(h^{m})\) for \(|\omega|<N\) and \(O(|\omega|^{-m})\) for \(|\omega|\geq N\).