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A note on measures whose diffraction is concentrated on a single sphere

Michael Baake, Jan Mazáč, Emily R. Korfanty

Abstract

Is there a translation-bounded measure whose diffraction is spherically symmetric and concentrated on a single sphere? This note constructively answers this question of Strungaru in the affirmative.

A note on measures whose diffraction is concentrated on a single sphere

Abstract

Is there a translation-bounded measure whose diffraction is spherically symmetric and concentrated on a single sphere? This note constructively answers this question of Strungaru in the affirmative.
Paper Structure (1 section, 2 theorems, 16 equations)

This paper contains 1 section, 2 theorems, 16 equations.

Table of Contents

  1. Acknowledgements

Key Result

Theorem 1

For any fixed $r>0$, the natural autocorrelation of the spherical wave $\space\mathrm{e}^{2 \pi \space\mathrm{i}\space r \| x \|}$ with $x\in\mathbb{R}\space^d$ exists, and is given by For $r\to 0^{+}$, this converges to the constant function $1$, so $\eta (x) \equiv 1$ and $\widehat{\eta} = \delta^{}_{0}$.

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Corollary 2