Expectations over deterministic fractal sets

摘要: Motivated by the need for new mathematical tools applicable to analysis of fractal point-cloud distributions, this thesis presents a measure-theoretic foundation consideration expectations smooth complex-valued functions over deterministic domains. Initial development theory proceeds from extension classical box integrals (pertaining separation moments unit hypercubes) special class sets known as String-generated Cantor Sets (SCSs). An experimental-mathematics approach facilitates discovery several closed-form results that indicate correct formulation fundamental definitions SCS sets. In particular, functional equations sets, supported underlying definitions, enable symbolic evaluation in cases (even-order or one-dimensional embeddings) and drive further developments theory, including establishment pole theorems, rationality construction high-precision algorithm general numerical computation expectations. The definition is subsequently generalised encompass all `deterministic' can be expressed attractor an Iterated Function System (IFS). This enables IFS attractors; Proposition 5.3.4. equation permits even-order attractors affine IFSs, such celebrated von Koch Snowflake Sierpinski Triangle. More generally, 5.3.4 provides means which any generated Collage Theorem order approximate digital image, Barnsley Fern, may symbolically resolved.