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Remote temperature sensing in 2D and the Bergman kernel

Steven R. Bell, Leah McNabb

Abstract

We explore the problem of estimating the steady state temperature in a two-dimensional domain at a point knowing the temperature to high order at another point. We find connections to the Bergman kernel of the domain, Runge's theorem, and approximate null quadrature identities.

Remote temperature sensing in 2D and the Bergman kernel

Abstract

We explore the problem of estimating the steady state temperature in a two-dimensional domain at a point knowing the temperature to high order at another point. We find connections to the Bergman kernel of the domain, Runge's theorem, and approximate null quadrature identities.

Paper Structure

This paper contains 5 sections, 1 theorem, 21 equations.

Table of Contents

  1. Introduction
  2. Proof of concept
  3. A more constructive approach
  4. Connections to Runge's theorem
  5. Further musings on the subject

Key Result

Theorem 4.1

Given a positive integer $N$, there is a positive integer $M$ and a constant $C>0$ such that, if the $C^M$-norm $\|H\|^M$ of $H$ on the outside of $\Omega$ is finite, then $h$ is in $C^N(\Omega)$ and the $C^N$-norm $\|h\|_N$ satisfies Consequently, $h(z)$ extends $C^\infty$-smoothly to the boundary of $\Omega$ from the inside if and only if $H(w)$ extends $C^\infty$-smoothly to the boundary of $\

Theorems & Definitions (1)

  • Theorem 4.1