The Squared Norm of the Error Functional for a Third-Derivative Interpolation Formula in a Hilbert Space

This work is devoted to the construction of an optimal interpolation formula for functions defined on a finite interval in the Hilbert space, which is characterized by the requirement that the third-order derivative of the function in this space needs to be absolutely continuous and integrable by the square of its fourth-order derivative. The interpolation formula under consideration is constructed so that it is exact both for polynomials up to the second degree and for an invariant exponential function, and the subspace covered by these functions plays a central role throughout the construction. The accuracy of the interpolation formula is characterized by its corresponding error functional, and the quality of the formula is measured by the norm of this function in the conjugate of the underlying Hilbert space. Based on the variational approach developed by S. L. Sobolev and his school for constructing optimal quadrature and interpolation formulas, an extremal function related to the error functional is introduced, and its exact expression is obtained by convolution with a special kernel function determined by the structure of the space. Using this extremal function together with the discrete orthogonality conditions satisfied by the interpolation formula, a precise closed-form expression for the square of the norm of the error functional is obtained in terms of the unknown coefficients of the formula. This expression serves as the main initial result on which the subsequent minimization problem is based to find the optimal coefficients of the interpolation formula.