### Abstract:

Subdivision is an important, iterative technique for the efficient generation of curves and surfaces in geometric modelling. The convergence of a subdivision scheme is closely connected to the existence of a corresponding refinable function. In turn, such a refinable fund ion can be used in the multi-resolutional construction method for wavelets, which are applied in many areas of signal analysis. As an introduction to subdivision, Chapter 1 gives a survey of some results in cornercutting subdivision for curves, and then the following Chapters 2 to 6 are devoted to the topic of interpolatory subdivision for curves First, in Chapter 2, after discussing Dubuc- Deslauriers subdivision, the author introduces a general class <i>A</i><sub>µ,υ</sub>, of symmetric interpolatory subdivision schemes with the property of polynomial filling up to a given odd degree 2υ- 1. Also, the author shows in Theorem 2.7 that any member of <i>A</i><sub>µ,υ</sub> is uniquely expressible in terms of a finite sequence of Dubuc- Deslauriers schemes. In Chapter 3, the author presents two construction methods for convergent schemes in <i>A</i><sub>µ,υ</sub>. The first method is based on the sampling at the half integers of a finitely supported function Q with appiopriate properties, and the second method uses a Bezout identity containing a llurwit.7. polynomial <i>H</i>. The author proceeds to develop, in Section 4.2, and as ultimately stated in Theorem 4.8, a sufficient condition, consisting of two inequalities, for convergence, which provides an alternative to a well-known existing condition due to C.A. Micchelli, and which is then shown, in Section 4.3, to be applicable for certain subclasses of <i>A</i><sub>µ,υ</sub>. Next, in Chapter 5, the author introduces, as an extension of Dubuc-Deslauriers subdivision, yet another construction method for schemes in <i>A</i><sub>µ,υ</sub>, as based on the sampling at ½ of a certain fundamental interpolant sequence, and for which, as stated in Corollary 5.7, an efficient computational method is then derived by using the Dubuc-Deslauriers expansion result of Theorem 2.7. In the setting of splines, the author then shows that her Theorem 4.8 yields convergence also for a subclass of subdivision schemes not satisfying the abovementioned Micchelli condition. All of the above subdivision schemes are based on the availability of a bi-infinite initial data sequence. Since, in many practical applications, a given finite initial data sequence can not be extended in a natural way to be bi-infinite, the author develops in Chapter 6, for the special case of Dubuc Deslauriers subdivision, a modified subdivision scheme which is equivalent to Dubuc Deslauriers subdivision away from the boundaries, and in such a way that the properties of interpolation and polynomial filling are preserved. Finally, in Chapter 7, the author uses the results of Chapter 6 to construct boundary-adapted interpolation wavelets, and then presents applications in signature smoothing and image decomposition.