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Box dimension of fractal interpolation surfaces with vertical scaling function

Lai Jiang

Abstract

In this paper, we first present a simple lemma which allows us to estimate the box dimension of graphs of given functions by the associated oscillation sums and oscillation vectors. Then we define vertical scaling matrices of generalized affine fractal interpolation surfaces (FISs). By using these matrices, we establish relationships between oscillation vectors of different levels, which enables us to obtain the box dimension of generalized affine FISs under certain constraints.

Box dimension of fractal interpolation surfaces with vertical scaling function

Abstract

In this paper, we first present a simple lemma which allows us to estimate the box dimension of graphs of given functions by the associated oscillation sums and oscillation vectors. Then we define vertical scaling matrices of generalized affine fractal interpolation surfaces (FISs). By using these matrices, we establish relationships between oscillation vectors of different levels, which enables us to obtain the box dimension of generalized affine FISs under certain constraints.
Paper Structure (9 sections, 15 theorems, 53 equations)

This paper contains 9 sections, 15 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition of fractal interpolation surfaces.
  4. Basic settings in this paper.
  5. Oscillation sums and box dimension.
  6. Vertical scaling matrices of generalized affine FISs
  7. Relationship between oscillations.
  8. Oscillation vectors and vertical scaling matrices.
  9. Calculation of box dimension of fractal interpolation surfaces

Key Result

Theorem 2.1

Let $\{ D \times {\mathbb R}, \Psi_{w}: w \in \Sigma \}$ be the IFS defined in eq:Wij-def. Then there exists a unique continuous function $f:\,D\to {\mathbb R}$ such that $f(x_i,y_j)=z_{ij}$ for all $0 \leq i \leq N$ and $0 \leq j \leq M$ and $\Gamma f=\bigcup_{w \in \Sigma} \Psi_w (\Gamma f)$, whe

Theorems & Definitions (25)

  • Theorem 2.1: MY10RX15
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 15 more