Some Analytical Properties of Multivariate Fractal Functions in Lebesgue Spaces

TL;DR

The paper develops multivariate fractal interpolation in Lebesgue spaces $L^q(\mathcal{I}^n, \mu_p)$ generated by an invariant measure $\mu_p$ from a multivariate IFS. It extends the Read-Bajraktarevič framework to these spaces, proving contraction of the RB operator on $L^q_0(\mathcal{I}^n, \mu_p)$ under $\|\alpha\|_\infty<1$, guaranteeing a fixed point $h^*\in L^q(\mathcal{I}^n, \mu_p)$. An $\alpha$-fractal operator $\mathcal{F}_{\Delta,L}^\alpha$ is defined with perturbation bound $\|f^\alpha-f\|_q \le \frac{\|\alpha\|_\infty}{1-\|\alpha\|_\infty}\|f-Lf\|_q$ and operator-norm bound $\|\mathcal{F}_{\Delta,L}^\alpha\|\le 1+\frac{\|\alpha\|_\infty\|Id-L\|}{1-\|\alpha\|_\infty}$. Under suitable bounds on $\|\alpha\|_\infty$, this operator is invertible (topological automorphism) or Fredholm with index zero, and the fixed points of $L$ and $\mathcal{F}_{\Delta,L}^\alpha$ coincide when $\|\alpha\|_\infty\neq 0$. Moreover, fractal functions $h_n^\alpha=\mathcal{F}_{\Delta,L}^\alpha(h_n)$ form a Schauder-basis-type family in $L^q(\mathcal{I}^n, \mu_p)$, enabling representations $g=\sum_n b_n h_n^\alpha$.

Abstract

In this article, we focus on the construction of multivariate fractal functions in Lebesgue spaces along with some properties of associated fractal operator. First, we give a detailed construction of the fractal functions belonging to Lebesgue spaces. Then, we give analytical properties of the defined fractal operator in Lebesgue spaces. We end this article by showing the existence of Schauder basis of the associated fractal functions for the space $\mathcal{L}^q(I^n, μ_p)$.

Figures (4)

• Figure 1: FIS($\alpha=0.3$)
• Figure 2: FIS($\alpha=0.5$)
• Figure 3: FIS($\alpha=0.7$)
• Figure 4: FIS($\alpha =0.9$)

Theorems & Definitions (12)

• Definition 1.1
• Theorem 1.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof