Optimal interpolation formulas in the Sobolev space of periodic functions W2(m)(T1)

The problem of finding the extremal function of an interpolation formula and of evaluating the norm of the error functional in the Sobolev space was first posed by S.L.Sobolev. In that work (Sobolev, 1961 [1] ), the extremal function of the interpolation formula was obtained explicitly in the Sobolev space whose generalized derivatives of order m are square-integrable. The present paper considers a new optimal interpolation formula in the Sobolev space: using the discrete analog of the differential operator, we obtain the optimal coefficients of the interpolation formula for an arbitrary m , and the norm of the error functional is computed. Integration of the interpolant leads to a quadrature formula; three numerical examples confirm the theoretical order of convergence.

Hali tarjima qilinmagan