System for finding the optimal coefficients of an interpolation spline

There are algebraic and variational approaches of construction in the spline theory. In algebraic approach splines are considered as some smooth piecewise polynomial functions. In the variational approach splines are elements of Hilbert or Banach spaces minimizing certain functionals. Then we study the problems of existence, uniqueness, and convergence of splines and algorithms for constructing them based on their own properties of splines. In this paper, we study the problem of constructing of interpolation spline in a Hilbert space. Here, we obtained a system of linear algebraic equations for the coefficients of the interpolation spline.

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