Optimal Difference Formulas in the Sobolev Space

Optimization of computational methods in functional spaces is one of the main problems of computational mathematics. In the present work algebraic and functional statements for the problem of difference formulas are discussed. For optimization of difference formulas, i.e., for construction of optimal difference formulas in functional spaces an important role is played by the extremal function of the given difference formula. In this work, in the Sobolev space, the extremal function of the difference formula is explicitly found and the norm of the error functional of the difference formula is calculated. Furthermore, existence and uniqueness of the optimal difference formula are proved.