Multivariate refinable functions with emphasis on box splines

Van der Bijl, Rinske (2008-03)

Thesis (MComm (Mathematics))--Stellenbosch University, 2008.

Thesis

The general purpose of this thesis is the analysis of multivariate refinement equations, with focus on the bivariate case. Since box splines are the main prototype of such equations (just like the cardinal B-splines in the univariate case), we make them our primary subject of discussion throughout. The first two chapters are indeed about the origin and definition of box splines, and try to elaborate on them in sufficient detail so as to build on them in all subsequent chapters, while providing many examples and graphical illustrations to make precise every aspect regarding box splines that will be mentioned. Multivariate refinement equations are ones that take on the form (x) =Xi2Zn pi (Mx − i), (1) where is a real-valued function, called a refinable function, on Rn, p = {pi}i2Zn is a sequence of real numbers, called a refinement mask, and M is an n × n matrix with integer entries, called a dilation matrix. It is important to note that any such equation is thus simultaneously determined by all three of , p and M — and the thesis will try and explain what role each of these plays in a refinement equation. In Chapter 3 we discuss the definition of refinement equations in more detail and elaborate on box splines as our first examples of refinable functions, also showing that one can actually use them to create even more such functions. Also observing from Chapter iii iv 2 that box splines demand yet another parameter from us, namely an initial direction matrix D, we focus on the more general instances of these in Chapter 4, while keeping the dilation matrix M fixed. Chapter 5 then in turn deals with the matrix M and tries to generalize some of the results found in Chapter 3 accordingly, keeping the initial direction matrix fixed. Having dealt with the refinement equation itself, we subsequently focus our attention on the support of a (bivariate) refinable function — that is, the part of the xy-grid on which such a function “lives” — and that of a refinement mask, in Chapter 6, and obtain a few results that are in a sense introductory to our work in the next chapter. Next, we move on to discuss one area in which refinable functions are especially applicable, namely subdivision, which is analyzed in Chapter 7. After giving the basic definitions of subdivision and subdivision convergence, and investigating the “sum rules” in Section 7.1, we prove our main subdivision convergence result in Section 7.2. The chapter is concluded with some examples in Section 7.3. The thesis is concluded, in Chapter 8, with a number of remarks on what has been done and issues that are left for future research.

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