Construction of optimal interpolation formula exact for trigonometric functions by Sobolev’s method

Abstract

The paper is devoted to derivation of the optimal interpolation formula in W2(0,2)(0,1) Hilbert space by Sobolev’s method. Here the interpolation formula consists of a linear combination ΣNβ=0Cβφ(xβ) of the given values of a function φ from the space W2(0,2)(0,1). The difference between functions and the interpolation formula is considered as a linear functional called the error functional. The error of the interpolation formula is estimated by the norm of the error functional. We obtain the optimal interpolation formula by minimizing the norm of the error functional by coefficients Cβ(z) of the interpolation formula. The obtained optimal interpolation formula is exact for trigonometric functions sinx and cosx. At the end of the paper we give some numerical results which confirm the numerical convergence of the optimal interpolation formula.

Keywords

• extremal function
• error functional
• hilbert space
• optimal interpolation formula
• optimal coefficients
• sobolev’s method
• extremal function
• error functional
• hilbert space
• optimal interpolation formula
• optimal coefficients
• sobolev’s method
• экстремальная функция
• функционал погрешности
• гильбертово пространство
• оптимальная интерполяционная формула
• оптимальные