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Identifying Bergman space functions from intervals

Andreas Hartmann, Marcu-Antone Orsoni

Abstract

We characterize functions of a Bergman space on a square by their values and derivatives on the diagonals. This problem is connected with the reachable space of the one-dimensional heat equation on a finite interval with boundary $L^2$-controls.

Identifying Bergman space functions from intervals

Abstract

We characterize functions of a Bergman space on a square by their values and derivatives on the diagonals. This problem is connected with the reachable space of the one-dimensional heat equation on a finite interval with boundary -controls.
Paper Structure (4 sections, 7 theorems, 84 equations, 4 figures)

This paper contains 4 sections, 7 theorems, 84 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Aikawa-Hayashi-Saitoh result
  3. The case of one diagonal
  4. Recovering the norm from two diagonals

Key Result

Theorem 1

For every function $f\in A^2(D)$, we have Moreover, if for a function $f$ in $\mathcal{C}^{\infty}((0,1)\cup (1/2-i/2,1/2+i/2))$ and holomorphic in a neighborhood of $1/2$, the right hand side of MainEstim is finite, then $f$ extends to $A^2(D)$, and the identity MainEstim holds.

Figures (4)

  • Figure 1: The domain $\Omega$.
  • Figure 2: The domain $\Omega$ and the ellipse $\operatorname{Ell}_{q,0}$.
  • Figure 3: The decomposition of $\Omega_q$.
  • Figure 4: The domains $\Omega$, $\Omega^a$ and $D$.

Theorems & Definitions (13)

  • Theorem
  • Theorem
  • proof
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 3 more