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The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators

Markus Faulhuber, Anupam Gumber, Irina Shafkulovska

Abstract

We study the spectral bounds of self-adjoint operators on the Hilbert space of square-integrable functions, arising from the representation theory of the Heisenberg group. Interestingly, starting either with the von Neumann lattice or the hexagonal lattice of density 2, the spectral bounds obey well-known arithmetic-geometric mean iterations. This follows from connections to Jacobi theta functions and Ramanujan's corresponding theories. As a consequence we re-discover that these operators resemble the identity operator as the density of the lattice grows. We also prove that the conjectural value of Landau's constant is obtained as the cubic arithmetic-geometric mean of $\sqrt[3]{2}$ and 1, which we believe to be a new result.

The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators

Abstract

We study the spectral bounds of self-adjoint operators on the Hilbert space of square-integrable functions, arising from the representation theory of the Heisenberg group. Interestingly, starting either with the von Neumann lattice or the hexagonal lattice of density 2, the spectral bounds obey well-known arithmetic-geometric mean iterations. This follows from connections to Jacobi theta functions and Ramanujan's corresponding theories. As a consequence we re-discover that these operators resemble the identity operator as the density of the lattice grows. We also prove that the conjectural value of Landau's constant is obtained as the cubic arithmetic-geometric mean of and 1, which we believe to be a new result.
Paper Structure (18 sections, 5 theorems, 118 equations, 4 figures)

This paper contains 18 sections, 5 theorems, 118 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Root systems and lattices
  4. Gabor systems over lattices
  5. AGMs and lattice theta functions
  6. Jacobi theta functions
  7. The analogues
  8. The AGM of theta functions
  9. Lattice theta functions and Gaussian lattice sums
  10. The fundamental identity of Gabor analysis
  11. Ramanujan's Corresponding Theories
  12. The constants G and L and proof of Equation
  13. Gabor Frame Bounds and the AGM
  14. Computing Gabor frame bounds
  15. The proof of Theorem \ref{['thm:agm2']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\Lambda_{1\times1}(2^{n}) = 2^{-n/2} \mathbb{Z} \times 2^{-n/2} \mathbb{Z}$, $n \in \mathbb{N}$, denote the scaled von Neumann lattice of density $2^n$. Denote the spectral bounds of the frame operator by $A_{1\times1}(2^n)$ and $B_{1\times1}(2^n)$. Then, they obey the arithmetic-geometric mean The constants $A_{1\times1}(2)$ and $B_{1\times1}(2)$ satisfy the relation $B_{1\times1}(2)/A_{1\ti

Figures (4)

  • Figure 1: The root systems $\mathsf{A}_1 \times \mathsf{A}_1$ and $\mathsf{D}_2$ are isomorphic as well as the root systems $\mathsf{B}_2$ and $\mathsf{C}_2$. All of them generate a (scaled) von Neumann lattice, by considering all integer linear combinations. The other existing root systems are $\mathsf{A}_2$ and $\mathsf{G}_2$. Both generate a hexagonal lattice. Note that $a A_1 \times b A_1$, $a, b > 0$ is also a root system which gives a rectangular lattice.
  • Figure 2: The von Neumann lattice $\mathbb{Z}^2$ contains the root system $\mathsf{A}_1 \times \mathsf{A}_1$. By adding new points in the center of the fundamental cell (deep hole) we obtain a lattice with twice the density. The original lattice is contained as a sub-lattice and the new lattice contains the root system $\mathsf{C}_2$. The new lattice is merely a rotation of the scaled original by 45 degrees.
  • Figure 3: The hexagonal lattice of density 1 contains the root system $\mathsf{A}_2$. By adding new points in the center of the fundamental triangle (deep hole) we obtain a lattice with thrice the density. The original lattice is contained as a sub-lattice and the new lattice contains the root system $\mathsf{G}_2$. The new lattice is merely a rotation of the scaled original by 30 degrees.
  • Figure 4: The lemniscate of Bernoulli and Erdös lemniscate of degree 3 (put to scale). The reflection of the respective lemniscate with respect to the unit circle (inverting distances) yields the corresponding hyperbolas. The intersections of their asymptotes with the unit circle yield root systems. These are generating a (scaled and rotated) von Neumann lattice and a hexagonal lattice, respectively.

Theorems & Definitions (10)

  • Theorem 1.1: AGM2
  • Theorem 1.2: AGM3
  • proof : Proof of \ref{['eq:Landau_ag3']}
  • Proposition 5.1: Janssen Jan96
  • proof
  • Theorem : Gauss, 1799
  • proof
  • Theorem : Borwein and Borwein, 1991
  • Conjecture 7.1: Strohmer, 2009
  • Conjecture 7.2