# On an Optimal Quadrature Formula in a Hilbert Space of Periodic Functions
###### Abstract  
The present work is devoted to the construction of optimal quadrature formulas for the approximate calculation of the integrals \(\int_{0}^{2\pi}\dot{e}^{\alpha\alpha}\varphi(x)dx\) in the Sobolev space \(\dot{H}_{2}^{m}\). Here, \(\dot{H}_{2}^{m}\) is the Hilbert space of periodic and complex-valued functions whose \(m\)-th generalized derivatives are square-integrable. Here, firstly, in order to obtain an upper bound for the error of the quadrature formula, the norm of the error functional is calculated. For this, the extremal function of the considered quadrature formula is used. By minimizing the norm of the error functional with respect to the coefficients, an optimal quadrature formula is then obtained. Using the explicit form of the optimal coefficients, the norm of the error functional of the optimal quadrature formula is calculated. The convergence of the constructed optimal quadrature formula is investigated, and it is shown that the rate of convergence of the optimal quadrature formula is \(O(h^{m})\) for \(|\omega|<N\) and \(O(|\omega|^{-m})\) for \(|\omega|\geq N\). Finally, we present numerical results of comparison for absolute errors of the optimal quadrature formula with the \(\exp(\mathrm{i}\omega x)\) weight in the case \(m=2\) and the Midpoint formula. There, one can see the advantage of the optimal quadrature formulas. Sobolev space of periodic functions; extremal function; error functional; optimal quadrature formulas; oscillating functions 202215344. [[https://doi.org/10.3390/a15100344](https://doi.org/10.3390/a15100344)]([https://doi.org/10.3390/a15100344](https://doi.org/10.3390/a15100344))  
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[PERSON], [PERSON] On an Optimal Quadrature Formula in a Hilbert Space of Periodic Functions. and the square of the norm for the error functional of the obtained optimal quadrature formula is evaluated in a certain Hilbert space. Suppose \(\widetilde{H}_{2}^{(m)},\;m\geq 1\) is the Hilbert space of \(2\pi\)-periodic, complex-valued functions \(\varphi(x),\;-\infty<x<\infty\), which are square-integrable with the \(m\)th generalized derivative. Every element of the space \(\widetilde{H}_{2}^{(m)}\) is a class of such functions that differ from each other by a constant term, and the inner product of the elements in this space is defined as  
\[(f,g)_{m}=\int\limits_{0}^{2\pi}f^{(m)}(x)\overline{g^{(m)}(x)}dx. \tag{2}\]  
Here, the notation \(g\) is the complex conjugate to \(g\). The norm of a function \(\varphi(x)\) in the space \(\widetilde{H}_{2}^{(m)}\) is defined by  
\[\left\|\varphi|\widetilde{H}_{2}^{(m)}\right\|=\left(\int\limits_{0}^{2\pi} \varphi^{(m)}(x)\overline{\varphi^{(m)}}(x)\right)^{\frac{1}{2}}. \tag{3}\]  
For \(\varphi(x)\in\widetilde{H}_{2}^{(m)}\), we consider a quadrature formula of the form  
\[\int\limits_{0}^{2\pi}\exp(\mathrm{i}\omega x)\varphi(x)dx\cong\sum\limits_{ k=1}^{N}C_{k}\varphi\bigg{(}\frac{2\pi k}{N}\bigg{)}, \tag{4}\]  
where \(C_{k}\) are coefficients of the quadrature formula, \(N=2,3,\ldots\). The error of the quadrature formula is given in the form  
\[(\ell,\varphi) = \int\limits_{0}^{2\pi}\exp(\mathrm{i}\omega x)\varphi(x)dx-\sum \limits_{k=1}^{N}C_{k}\varphi\bigg{(}\frac{2\pi k}{N}\bigg{)} \tag{5}\] \[= \int\limits_{0}^{2\pi}\bigg{[}\exp(\mathrm{i}\omega x)-\sum \limits_{k=1}^{N}C_{k}\sum\limits_{\beta=-\infty}^{\infty}\delta\bigg{(}x- \frac{2\pi k}{N}-2\pi\beta\bigg{)}\bigg{]}\varphi(x)dx.\]  
Here, \(\delta(x)\) is the Dirac delta-function,  
\[\ell(x)=\exp(\mathrm{i}\omega x)-\sum\limits_{k=1}^{N}C_{k}\sum\limits_{\beta =-\infty}^{\infty}\delta\bigg{(}x-\frac{2\pi k}{N}-2\pi\beta\bigg{)}, \tag{6}\]  
\(\ell\) is the periodic functional of the error for the quadrature formula, and \((\ell,\varphi)\) is the value of the error functional at \(\varphi\).