On one modification of the Sobolev method for approximating a function
Keywords:
Hilbert space, extremal function, error functional, interpolation formulaAbstract
In spline theory, there are algebraic and variational approaches. In the algebraic approach, splines are considered as some smooth piecewise polynomial functions. In the variational approach, splines are understood as elements of a Hilbert or Banach space that minimize certain functionals. Then, the problems of existence, uniqueness, and convergence of splines and algorithms for constructing them based on the proper properties of splines are studied. In this paper, we study the problem of constructing optimal interpolation formulas in a Hilbert space. Here, using the Sobolev method, an algorithm is given for solving a system of linear algebraic equations for the coefficients of optimal interpolation formulas. An explicit expression for the optimal coefficients of an interpolation formula in a Hilbert space is obtained.
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