Interpolation and approximation by q-B-spline functions

In this thesis, from viewpoint of interpolation and approximation, we investigate fundamental properties of q-B-spline functions which are piecewise polynomials whose q-derivatives are continuous. We give a representation of q-B-splines via divided differences of truncated power functions which may be considered as a generalization of classical truncated functions. We derive q-analogue of the de Boor-Fix formula for dual functions and 1/q-integral of q-B-splines over their support by using recurrence formula for q-B-splines and q-analogue of Marsden identity. A 1/q-convolution formula for uniform q-B-splines is also obtained. We introduce a q-analogue of Peano kernel theorem. We give applications to polynomial interpolation and construct examples in which classical remainder theory fails whereas q-Peano kernel works. Furthermore, we establish a relation between q-B-splines and divided differences via q-Peano kernel. We show that q-B-splines form a basis for quantum spline spaces. Recurrence relations for 1/q-integration and 1/q-differentiation formulas of q-spline functions are derived. Moreover, we demonstrate a way to find the polynomials on each interval of a q-spline function. After all, we give basic theory of interpolation by a q-B-splines and approximation by q-analogue of the Bernstein-Schoenberg operator.