On 28 August 2019 at 11.15 Gul Wali Shah will defend his doctoral thesis „Spline approximations“.
Senior Research Fellow Peeter Oja, University of Tartu
Lecturer Evely Kirsiaed, University of Tartu
Associate professor, Svetlana Asmuss, University of Latvia
Professor Jaan Janno, Tallinn University of Technology
The dissertation treats three kinds of problems from the theory of splines. Firstly, a par¬ti-cular interpolation problem about the cubic spline histopolation with arbitrary place¬ment of histogram knots and spline knots between them is discussed. A cubic spline is studied provided that its integral on a prescribed interval equals the area of the corres¬pon¬ding histogram rectangle. It is considered the most common boundary value con¬di¬tions like given values of the spline and its first and second derivatives in endpoints of given interval and then solved the problem of existence and uniqueness of the solution for such histopolation problem. Secondly, the periodic polynomial spline histopolation problem with the arbitrary placement of histogram knots and coinciding histogram knots is considered. Several results about the existence and uniqueness of solution are obtained and they imply known results in the case of uniform grid. In the last problem, the rational spline histopolation of convex data is studied. For the concern about the con¬vexity, an appropriate tool is interpolation or histopolation with quadratic/linear rational splines because these splines keep the sign of its second derivative on the whole interval. For this reason the given histogram is assumed to be strictly convex. The main task is at the study of existence of solution for a nonlinear system of basic equations to determine the values of second derivatives in spline knots. The other parameters in the representation of spline are determined from a linear system with regular matrix. It is shown that there is a strictly convex histogram without the solution of histopolation problem for any choice of spline knots.