Pahiria M. Generalization of classical continued fractions and function approximation.

The problem of interpolation functions by continued fraction, the problem of expansion functions in continued fractions and the problem of interpolation functional by integral continued C-fraction are investigated in the thesis. New estimates for the remainders of the functional continued fractions of a complex variable and interpolation continued fractions of a real variable with polynomial elements have been proved. Problems of approximation of functions by Thiele interpolation continued fraction and interpolation continued C--fraction have been investigated. Estimates of the remainders of the interpolation continued fractions of the functions of the complex variable have been obtained, the convergence of interpolation processes have been proved. The problem of interpolation of a functional by an integral continued C-fraction if its value is known on the set of continual nodes has been studied. The necessary and sufficient conditions for its solvability have been obtained. The quasi--reciprocal interpolation continued fractions of Thiele type and C--fractions type have been considered. Estimates of remainders of interpolation continued fractions following types have been received, the convergence of interpolation processes has been proved. The new type of reciprocal differences -- reciprocal differences of the 2nd type have been introduced, their properties have been proved. Functional interpolation continued fractions and quasi--reciprocal functional interpolation continued fractions have been proposed for the first time. Estimates of the remainders of functional interpolation continued fractions and convergence of interpolation processes have been proved. The reciprocal g-difference and the reciprocal g--difference of the 2nd type have been introduced. Properties of reciprocal g-differences and reciprocal g-difference of the 2nd type have been proved. New properties of Thiele reciprocal derivatives have been obtained, rules of reciprocal differentiation by Thiele have been established. The equivalence of different ways of expansion a function in a regular C-fraction has been proved. Areas of convergence of expansion of some functions in continued fraction, a priori and a posteriori estimates, have been established. For the first time the reciprocal derivatives of the 2nd type have been introduced, the properties of such a type of reciprocal derivatives and Thiele type formula have been obtained. Relationships between the reciprocal derivatives of Thiele and the ordinary derivatives of the function have been established. The possibility of expansion of a function in a quasi--reciprocal continued C--fraction, convergence of such expansion to function, a posteriori estimation have been proved. A new notations -- an reciprocal g--derivative and reciprocal g--derivative of the 2nd type have been introduced. Properties of such reciprocal derivatives and functional formulas of the Thiele type have been proved.