The paper studies the problem of construction of optimal interpolation
formulas with derivative in the Sobolev space L(m)2 (0,1). Here the
interpolation formula consists of the linear combination of values of the
function at nodes and values of the first derivative of that function at the
end points of the interval [0,1]. For any function of the space L(m)2 (0,
1) the error of the interpolation formulas is estimated by the norm of the
error functional in the conjugate space L(m)* 2 (0,1). For this, the norm
of the error functional is calculated. Further, in order to find the minimum
of the norm of the error functional, the Lagrange method is applied and the
system of linear equations for coefficients of optimal interpolation formulas
is obtained. It is shown that the order of convergence of the obtained
optimal interpolation formulas in the space L(m)2 (0,1) is O(hm). In order
to solve the obtained system it is suggested to use the Sobolev method which
is based on the discrete analog of the differential operator d2m= dx2m. Using
this method in the cases m = 2 and m = 3 the optimal interpolation formulas
are constructed. It is proved that the order of convergence of the optimal
interpolation formula in the case m = 2 for functions of the space C4(0,1)
is O(h4) while for functions of the space L(2)2 (0,1) is O(h2). Finally,
some numerical results are presented.
An optimal interpolation formula
