Approximation of non-analytic and/or highly oscillatory functions via their taylor expansion whose remainder term is expressed in integral form under a weight function and development of numerical integration methods related to this method

Approximate value calculations, function approximations and quadratures, either because of structural reasons or because they constitute a basis for natural sciences, have always been a center of attention for mathematicians. In this thesis Taylor expansion which has an important place in the approximation theory and which makes an important starting point for the construction of many new methods, is taken into consideration and some more productive methods are searched by working on the remainder term expressed in integral form In the first part of the thesis a short introduction is given, approximations are classified and the basic philosophies of interpolation and curve fitting are mentioned. In the second part of the thesis Fluctuation Theorem, its application to Taylor expansion, the reorganisation of Taylor expansion to make room for this application and finally theorems and applications concerning EMPR are given in detail. In the third part new research and findings are given and discussed in details. It starts with fluctuation studies on the matrix representation of operator multiplication and decomposition. Then the approximation to highly oscillatory functions by applying trigonometric basis functions to the remainder term of Taylor expansion. This is followed by the application of EMPR to the remainder term of Taylor expansion expressed as a contour integration and the application of Fluctuationlessness Theorem on contour integration with circular contours. Then using TKEMPR on the remainder term of Taylor expansion expressed in contour integral form is taken into consideration. Finally Euler and higher order Taylor methods for initial value problems are worked on and beter results are reached through fluctuationlessness approximation applied on them. In the fourth section some corollaries are listed.