[Paper Review] Approximation in $AC(σ)$

Ian Doust, Michael Leinert+1 more|UNSWorks (University of New South Wales, Sydney, Australia)|Dec 6, 2013

Advanced Banach Space Theory9 references3 citations

TL;DR

This paper establishes that in the space $AC(\sigma)$ of absolutely continuous functions on a compact subset $\sigma \subset \mathbb{R}^2$, the spaces of continuously differentiable functions $C^1(\sigma)$ and continuous piecewise planar functions $\mathrm{CTPP}(\sigma)$ are dense. The authors prove this via approximation by polynomials and piecewise planar functions using variation norms, extending classical results from $AC[0,1]$ to the two-dimensional setting.

ABSTRACT

For a nonempty compact subset $σ$ in the plane, the space $AC(σ)$ is the closure of the space of complex polynomials in two real variables under a particular variation norm. In the classical setting, $AC[0,1]$ contains several other useful dense subsets, such as continuous piecewise linear functions, $C^1$ functions and Lipschitz functions. In this paper we examine analogues of these results in this more general setting.

Motivation & Objective

• To extend classical approximation results from $AC[0,1]$ to the multivariable setting of $AC(\sigma)$ for compact $\sigma\subset\mathbb{R}^2$.
• To investigate whether standard dense subsets like $C^1(\sigma)$, Lipschitz functions, and continuous piecewise linear functions have analogues in $AC(\sigma)$.
• To establish that $C^1(\sigma)$ and $\mathrm{CTPP}(\sigma)$ are dense in $AC(\sigma)$ under the variation norm.

Proposed method

• Define the variation norm $\|f\|_{BV(\sigma)} = \|f\|_\infty + \operatorname{var}(f,\sigma)$, where $\operatorname{var}(f,\sigma)$ is the supremum of $\operatorname{cvar}(f,S)/\operatorname{vf}(S)$ over finite lists $S\subset\sigma$.
• Introduce $\mathrm{CTPP}(\sigma)$, the space of continuous piecewise planar functions on $\sigma$, as a two-dimensional analogue of piecewise linear functions on $[0,1]$.
• Use the local nature of absolute continuity to reduce global approximation to local approximation on compact neighborhoods.
• Apply the Mean Value Theorem and Rolle’s Theorem on triangles to bound the difference between a $C^1$ function and its piecewise planar interpolant.
• Leverage the fact that polynomials are dense in $AC(\sigma)$ and use approximation by piecewise planar functions to show density of $\mathrm{CTPP}(\sigma)$ and $C^1(\sigma)$.
• Extend results from real-valued to complex-valued functions by separating real and imaginary parts and applying the same approximation techniques.

Experimental results

Research questions

• RQ1 Can $C^1(\sigma)$ functions be approximated in the $BV(\sigma)$-norm by simpler functions in $AC(\sigma)$ for compact $\sigma\subset\mathbb{R}^2$?
• RQ2 Is there a two-dimensional analogue of continuous piecewise linear functions that is dense in $AC(\sigma)$?
• RQ3 Do the classical approximation results for $AC[0,1]$ extend to the multivariable setting of $AC(\sigma)$?
• RQ4 Can $C^1(\sigma)$ functions be approximated by piecewise planar functions in the variation norm?
• RQ5 Are $\mathrm{CTPP}(\sigma)$ and $C^1(\sigma)$ dense in $AC(\sigma)$ for complex-valued functions on $\sigma$?

Key findings

• The space $C^1(\sigma)$ is dense in $AC(\sigma)$ under the variation norm, extending the classical result from $AC[0,1]$ to compact planar sets.
• The space $\mathrm{CTPP}(\sigma)$ of continuous piecewise planar functions is dense in $AC(\sigma)$, providing a natural multivariable analogue of piecewise linear approximation.
• For any $f\in AC(\sigma)$ and $\varepsilon>0$, there exists $g\in\mathrm{CTPP}(\sigma)$ such that $\|f-g\|_{BV(\sigma)}<\varepsilon$, with explicit bounds derived from local approximation on triangles.
• The same approximation result holds for complex-valued functions in $AC(\sigma)$, as both real and imaginary parts can be approximated by $\mathrm{CTPP}(\sigma)$ functions.
• The variation norm allows for a local-to-global argument: if a function is absolutely continuous on a neighborhood of each point in $\sigma$, then it is in $AC(\sigma)$.
• The density of $C^1(\sigma)$ in $AC(\sigma)$ follows from the fact that polynomials are dense in $AC(\sigma)$ and $C^1(\sigma)$ contains all polynomials.