Construction of interpolation splines minimizing semi-norm in W_{2}^{(m,m-1)}(0,1) space

In the present paper using S.L. Sobolev’s method interpolation splines minimizing the semi-norm in a Hilbert space are constructed. Explicit formulas for coefficients of interpolation splines are obtained. The obtained interpolation spline is exact for polynomials of degree m?2 and e ^?x . Also some numerical results are presented.     

Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space $L_{2}^{(m)}(0,1)$

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the \(L_{2}^{(m)}(0,1)\) space for numerical calculation of Fourier coefficients. Using the S.L.Sobolev’s method, we obtain new optimal quadrature formulas of such type for N+1≥m, where N+1 is the number of nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formula for the case m=1. The obtained optimal quadrature formulas in the \(L_{2}^{(m)}(0,1)\) space are exact for P _m?1(x), where P _m?1(x) is a polynomial of degree m?1. Furthermore, we present some numerical results, which confirm the obtained theoretical results.     

Optimal quadrature formulas for Fourier coefficients in $W_2^{(m,m-1)}$ space

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the $W_2^{(m,m-1)}[0,1]$ space for calculating Fourier coefficients. Using S.~L.\ Sobolev''s method we obtain new optimal quadrature formulas of such type for $N 1\geq m$, where $N 1$ is the number of the nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We investigate the order of convergence of the optimal formula for $m=1$. The obtained optimal quadrature formula in the $W_2^{(m,m-1)}[0,1]$ space is exact for $\exp(-x)$ and $P_{m-2}(x)$, where $P_{m-2}(x)$ is a polynomial of degree $m-2$. Furthermore, we present some numerical results, which confirm the obtained theoretical results.     

The discrete analogue of a differential operator and its applications

We construct a discrete analogue D _m (hβ) of the differential operator d^2m /dx ^2m + 2ω ²d^2m?2 /dx ^2m?2 + ω ⁴d^2m?4 /dx ^2m?4 for any m ≥ 2. In the case m = 2, we apply in the Hilbert space K ₂(P ₂) the discrete analogue D ₂(hβ) for construction of optimal quadrature formulas and interpolation splines minimizing the seminorm, which are exact for trigonometric functions sin ωx and cos ωx.     

On an optimal quadrature formula in the sense of Sard

In this paper we construct an optimal quadrature formula in the sense of Sard in the Hilbert space K ₂(P ₂). Using S.L. Sobolev’s method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order of the convergence of the optimal formula and prove an asymptotic optimality of such a formula in the Sobolev space L₂⁽²⁾(0,1)L_2^{(2)}(0,1). The obtained optimal quadrature formula is exact for the trigonometric functions sinx and cosx. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.