Interpolation of functionals by integral continued C-fractions

Abstract

The functional interpolation problem on a continual set of nodes by an integral continued C-fraction is studied. The necessary and sufficient conditions for its solvability are found. As a particular case, the considered integral continued fraction contains a standard interpolation continued C-fraction which is used to approximate the functions of one variable.

Interpolation integral continued fraction with twofold node

“…Let us check the interpolation conditions. Interpolativity in the node x 1 (·, ξ 1 ) = x 0 (·) + (x 1 (·) − x 0 (·))H(· − ξ 1 ) follows from [4] because q E 2 (x 1 (·, ξ 1 )) = 0. Now we will check interpolativity of the derivatives.…”

“…where 1. q E l (x(·)) = q l (x(·)) for l k; Proof. According to [4] equality Q m+1 x m+1 (·, ξ m+1 ) = F x m+1 (·, ξ m+1 ) takes place. Taking into consideration (11) we have that Q m+1 x m+1 (·, ξ m+1 ) = F x m+1 (·, ξ m+1 ) .…”

“…Then the interpolation ICF will not transform in a interpolation continued fraction for a function of a single variable. To solve this problem, in [4] the new class of the interpolation ICF of the following type has been introduced Q n (x(·)) =…”

“…. In [4] it is proved that the necessary condition for interpolativity of ICF (2) for a smooth functional F (x(·)) : Q[0, 1] → P 1 on continual nodes set (1) is following: its kernel must be defined by formulas…”