arXiv subjects

A. R. Hayotov

Publications and source records attributed to A. R. Hayotov.

14 recordsLinked to original sources

arXiv2021

Optimal quadrature formulas for computing of Fourier integrals in a Hilbert space

In the present paper the optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral $\int_a^b e^{2πiωx}φ(x)d x$ with $ω\in \mathbb{R}$ in the Hilbert space $W_2^{(2,1)}[a,b]$ of complex-valued functions. Furthermore, the explicit expressions for coefficients of the constructed optimal quadrature formulas are obtained. At the end of the paper some numerical results are presented.

math.NA↗

arXiv2021

On an optimal quadrature formula for approximation of Fourier integrals in the space $W_2^{(1,0)}$

The present paper is devoted to construction of an optimal quadrature formula for approximation of Fourier integrals in the Hilbert space $W_2^{(1,0)}[a,b]$ of non-periodic, complex valued functions. Here the quadrature sum consists of linear combination of the given function values on uniform grid. The difference between integral and quadrature sum is estimated by the norm of the error functional. The optimal quadrature formula is obtained by minimizing the norm of the error functional with respect to coefficients. In addition, analytic formulas for optimal coefficients are obtained using the discrete analogue of the differential operator $d^2/d x^2-1$. Further, the order of convergence of the optimal quadrature formula is studied.

math.NA↗

arXiv2019

Construction of optimal quadrature formulas exact for exponentional-trigonometric functions by Sobolev's method

The paper studies Sard's problem on construction of optimal quadrature formulas in the space $W_2^{(m,0)}$ by Sobolev's method. This problem consists of two parts: first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas. Here the norm of the error functional is calculated with the help of the extremal function. Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space $W_2^{(m,0)}$ is obtained, moreover the existence and uniqueness of the solution of this system are discussed. Next, the discrete analogue $D_m(hβ)$ of the differential operator $\frac{d^{2m}}{d x^{2m}}-1$ is constructed. Further, Sobolev's method of construction of optimal quadrature formulas in the space $W_2^{(m,0)}$, which based on the discrete analogue $D_m(hβ)$, is described. Finally, for $m=1$ and $m=3$ the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained.

math.NA↗

arXiv2018

Optimal interpolation formulas in $W_2^{(m,m-1)}$ space

In the present paper optimal interpolation formulas are constructed in $W_2^{(m,m-1)}(0,1)$ space. Explicit formulas for coefficients of optimal interpolation formulas are obtained. Some numerical results are presented.

math.NA↗

arXiv2014

Optimal quadrature formulas with derivatives in Sobolev space

In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space $L_2^{(m)}(0,1)$is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first and the third derivatives of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number $N$ and for any $m\geq 4$ using S.L. Sobolev method which is based on discrete analogue of the differential operator $d^{2m}/dx^{2m}$. In particular, for $m=4,\ 5$ optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from $m=6$ new optimal quadrature formulas are obtained.

math.NA↗

arXiv2013

The discrete analogue of the differential operator $\frac{d^{2m}}{d x^{2m}} + 2ω^2\frac{d^{2m-2}}{d x^{2m-2}} + ω^4\frac{d^{2m-4}}{d x^{2m-4}}$

In the present paper we construct the discrete analogue $D_m(hβ)$ of the differential operator $\frac{d^{2m}}{d x^{2m}} + 2ω^2\frac{d^{2m-2}}{d x^{2m-2}} + ω^4\frac{d^{2m-4}}{d x^{2m-4}}$. The discrete analogue $D_m(hβ)$ plays the main role in construction of optimal quadrature formulas and interpolation splines minimizing the semi-norm in the $K_2(P_m)$ Hilbert space.

math.NA↗

arXiv2012

Construction of discrete analogue of the differential operator $\frac{d^4}{dx^4}+2\frac{d^2}{dx^2}+1$ and its properties

In the present paper the discrete analogue of the differential operator $\frac{d^4}{dx^4}+2\frac{d^2}{dx^2}+1$ is constructed and its some properties are proved.

math.NA↗

arXiv2010

Optimal interpolation formulas in the periodic function space of S.L. Sobolev

In this paper the problem of construction of lattice optimal interpolation formulas in the space $\widetilde{L_2^{(m)}} (0,1)$ is considered. Using S.L. Sobolev's method explicit formulas for the coefficients of lattice optimal interpolation formulas are given and the norm of the error functional of lattice optimal interpolation formulas is calculated. Moreover, connection between optimal interpolation formula in the space $\widetilde{L_2^{(m)}} (0,1)$ and optimal quadrature formula in this space is shown. Finally, numerical results are given.

math.NA↗

arXiv2010

Optimal Quadrature Formulas in the Sense of Sard in $W_2^{(m,m-1)}$ Space

In this paper in $W_2^{(m,m-1)}(0,1)$ space the problem of construction of optimal quadrature formula in the sense of Sard is considered and using S.L. Sobolev's method it is obtained new optimal quadrature formula of such type. For the optimal coefficients explicit formulas are obtained. Furthermore, the numerical results which confirm the theoretical results of this work is given.

math.NA↗

arXiv2009

Optimal Quadrature Formulas with Positive Coefficients in $L_2^{(m)}(0,1)$ Space

In the Sobolev space $L_2^{(m)}(0,1)$ optimal quadrature formulas with the nodes (1.5) are investigated. For optimal coefficients explicit form are obtained and norm of the error functional is calculated. In particular, by choosing parameter $η_0$ in (1.5) the optimal quadrature formulas with positive coefficients are obtained and compared with well known optimal formulas.

math.NA↗

arXiv2008

On an optimal quadrature formula in Sobolev space $L_2^{(m)} (0,1)$

In this paper in the space $L_2^{(m)}(0,1)$ the problem of construction of optimal quadrature formulas is considered. Here the quadrature sum consists on values of integrand at nodes and values of first derivative of integrand at the end points of integration interval. The optimal coefficients are found and norm of the error functional is calculated for arbitrary fixed $N$ and for any $m\geq 2$. It is shown that when $m=2$ and $m=3$ the Euler-Maclaurin quadrature formula is optimal.

math.NA↗

arXiv2008

Calculation of the Norm of the Error Functional of Optimal Quadrature Formulas in the Space $W_2^{(2,1)}(0,1)$

In this paper in the space $W_2^{(2,1)}(0,1)$ square of the norm of the error functional of a optimal quadrature formula is calculated.

math.NA↗

arXiv2008

Calculation of Coefficients of the Optimal Quadrature Formulas in the $W_2^{m,m-1}(0,1)$ Space

In this paper problem of construction of optimal quadrature formulas in $W_2^{(m,m-1)}(0,1)$ space is considered. Here by using Sobolev's algorithm when $m=1,2$ we find the optimal coefficients of the quadrature formulas of the form $$ \int\limits_0^1ϕ(x)dx\cong \sum\limits_{β=0}^NC_βϕ(x_β). $$

math.NA↗

arXiv2008

Properties of Discrete Analogue of the Differential Operator $\frac{d^{2m}}{dx^{2m}}-\frac{d^{2m-2}}{dx^{2m-2}}$

In the paper properties of the discrete analogue $D_m(hβ)$ of the differential operator $\frac{d^{2m}}{dx^{2m}}-\frac{d^{2m-2}}{dx^{2m-2}}$ are studied. It is known, that zeros of differential operator $\frac{d^{2m}}{dx^{2m}}-\frac{d^{2m-2}}{dx^{2m-2}}$ are functions $e^x$, $e^{-x}$ and $P_{2m-3}(x)$. It is proved that discrete analogue $D_m(hβ)$ of this differential operator also have similar properties.

math.NA↗