الفهرس 
Chapter 1 IntroductionOnly 14 pages are availabe for public view
 |  | from | 76 |  |  |
| | | from | 76 | | |
Abstract
”This thesis belongs to the field of mathematical analysis, especially convex analysis. Convexity is a topic that has been studied in mathematics for a long time. Herman Minkowski defined convexity for the first time in 1896. The subject of the present work dealing with the issue of filling the large gap between the bounded Hӧlder function and its envelope so the convex envelope of the function from below is a poor approximation. For this purpose, the general (lower, upper, and mixed) compensated convex transformations (For short, CCTs) is a tight approximation method of functions. The importance of these transformations appears in many fields such as singularity extraction, image processing, and interpolation. Therefore, the locality property has been studied for compensated convex transforms. Also, the radius of locality for functions in the Hӧlder space has been presented. Also, we studied the Hausdorff stability for the q-upper compensated convex transformations, by showing that the q-upper compensated convex transformation is Hausdorff-Lipschitz continuous concerning closed sample sets at Hӧlder function f. This thesis contains several remarks and illustrative examples as an application of our results. The thesis is organized as follows: Chapter one deals with an introductory chapter, containing some preliminaries, definitions, basic concepts, elementary results of convex analysis, some basics of mathematical analysis, and compensated convexity that will be used throughout the next chapters. In chapter two, the locality property has been studied for compensated convex transforms. Also, the radius of locality for functions in the Hӧlder space has been presented. In chapter three, we studied the Hausdorff stability for the q-upper compensated convex transformations, by showing that the q-upper compensated convex transformation is Hausdorff-Lipschitz continuous concerning closed sample sets at Hӧlder function f. ”