Polinom olmayan B-spline fonksiyonları

This thesis is concerned with the explicit representation of non-polynomial B-spline functions for a wide collection of spline spaces including trigonometric splines, hyperbolic splines, and special Müntz spaces of splines. Divided differences are a basic tool in interpolation and approximation by polynomials and in spline theory. They are directly involved in the definition of B-splines. As in the original definition of B-splines, we represent the non-polynomial B-splines by using non-polynomial divided differences applied to a proper generalization of truncated-power function. Starting with the interpolation problem, we define the non-polynomial divided differences recursively as a generalization of classical divided differences. We also derive the identities and the properties of these non-polynomial divided differences such as symmetry, Leibniz formula, cancellation formula, affine combinations, and ratio of determinants. With the definition of a generalized derivative operator, it is shown that as in the polynomial case, non-polynomial divided differences can be viewed as a discrete analogue of derivatives. A generalization of the Hermite interpolation is also obtained using non-polynomial divided differences with repeated points. Using the link between non-polynomial divided differences and derivatives, we obtain the generalized Taylor series and state Generalized Taylor Theorem. With the definition of definite integral, the non-polynomial divided difference is built by the integration with non-polynomial B-spline functions. Also, we derive a general form of the Peano kernel theorem based on a generalized Taylor expansion. Here, we use the generalized Taylor Theorem with the integral remainder. As in the polynomial case, it is shown that the non-polynomial B-splines are in fact the Peano kernels of non-polynomial divided differences.