Bubniak M. Sets of Convergence for Periodic Branched Continued Fractions of the Special Form.

In the thesis the periodic branched continues fractions of the special form have been defined, the formula of the difference between their approximants, the formula between the approximant of p -periodic branched continued fractions and its value, that is expressed by fixed points of the respective linear fraction transformations, have been established.The multidimentional criteria of the pointwise or uniform convergence of these fractions that are mulditimensional analogs of the famous criteria of convergence of continued fractions have been proved. The pointwise convergence was established for 1-periodic branched continued fractions under some conditions: if partial nominator of the first branch belongs to the plain with a cut and all other elements belong to circle regions, or unions parabolar regions, or the plains with cuts. Angle regions and necessary condition of convergence are researched for 1-periodic branched continued fractions of the special form.The multidimentional generalizations of the oval theorem for the -periodic branched continued fractions have been proved. They have been applied for increasing the circle regions. The oval regions of uniform convergence for periodic branched continued fractions have been investigated for the first time. The truncation errors bounds have been investigated for the p-periodic branched continued fractions of the special form if their elements belong to the oval regions, which satisfy some additional conditions