Thesis supervisor: Peeter Oja, Assoiate Professor, University of Tartu
Opponents: Natalja Budkina, Dr. math., Riga Technical University
Jaan Janno, professor, Tallinn University of Technology
The data needed to use in data analysis is often given as an histogram, i.e., avarages of some variable on known (time)interval. For further analysis it is important them to be continious. To do that several methods can be used. The most used method is the interpolation, the histopolation has not been used so often. Usually the preservation of geometrical properties of given data like positivity (nonnegativity), monotonicity, convexity, etc., is needed. It is generally known that interpolation and histopolation with polynomial splines do not preserve geometrical properties of data.
The main research task in this thesis is to study the shape-preserving rational spline histopolation. Combined splines using linear/linear rational functions and quadratic polynomials on corresponding subintervals for preservation of monotonicity and quadratic/linear rational splines to preserve the convexity is studied. For any data, srategies to preserve monotonicity on maximal possible number of intervals are investigated. For actual construction of the histopolant a system of nonlinear equations has to be solved. Newton's method and ordinary iteration method are convenient for that. The convergence rate of such combined comonotone histopolating splines is established. An important and difficult point in reserch is finding sufficient conditions for the existence of convexity preserving histopolating rational spline. Several numerical exsample and illustrative figures are presented.