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Exponential bases for parallelepipeds with frequencies lying in a prescribed lattice

Dae Gwan Lee, Goetz E. Pfander, David Walnut

Abstract

The existence of a Fourier basis with frequencies in $\mathbb{R}^d$ for the space of square integrable functions supported on a given parallelepiped in $\mathbb{R}^d$, has been well understood since the 1950s. In a companion paper, we derived necessary and sufficient conditions for a parallelepiped in $\mathbb{R}^d$ to permit an orthogonal basis of exponentials with frequencies constrained to be a subset of a prescribed lattice in $\mathbb{R}^d$, a restriction relevant in many applications. In this paper, we investigate analogous conditions for parallelepipeds that permit a Riesz basis of exponentials with the same constraints on the frequencies. We provide a sufficient condition on the parallelepiped for the Riesz basis case which directly extends one of the necessary and sufficient conditions obtained in the orthogonal basis case. We also provide a sufficient condition which constrains the spectral norm of the matrix generating the parallelepiped, instead of constraining the structure of the matrix.

Exponential bases for parallelepipeds with frequencies lying in a prescribed lattice

Abstract

The existence of a Fourier basis with frequencies in for the space of square integrable functions supported on a given parallelepiped in , has been well understood since the 1950s. In a companion paper, we derived necessary and sufficient conditions for a parallelepiped in to permit an orthogonal basis of exponentials with frequencies constrained to be a subset of a prescribed lattice in , a restriction relevant in many applications. In this paper, we investigate analogous conditions for parallelepipeds that permit a Riesz basis of exponentials with the same constraints on the frequencies. We provide a sufficient condition on the parallelepiped for the Riesz basis case which directly extends one of the necessary and sufficient conditions obtained in the orthogonal basis case. We also provide a sufficient condition which constrains the spectral norm of the matrix generating the parallelepiped, instead of constraining the structure of the matrix.
Paper Structure (16 sections, 20 theorems, 58 equations, 4 figures)

This paper contains 16 sections, 20 theorems, 58 equations, 4 figures.

Table of Contents

  1. Introduction and main results
  2. Related work
  3. Organization of the paper
  4. Preliminaries
  5. Riesz Bases and perturbation theorems
  6. Density of frequency sets and invariance of exponential Riesz bases
  7. Lattices and Fuglede's theorem
  8. The Fourier transform and Paley--Wiener spaces
  9. Proof of the equivalence of Theorems \ref{['thm:main2-RB-general']} and \ref{['thm:main2-RB-integer']}
  10. Proof of Theorem \ref{['thm:main2-RB-integer']}
  11. Proof of Proposition \ref{['prop:main2-lower-triangular']} for $d = 2$
  12. Proof of Proposition \ref{['prop:main2-lower-triangular']} for $d \geq 2$
  13. Proof of Theorem \ref{['thm:limiting-spectral-norm']}
  14. Proof of Theorem \ref{['thm:tensor-product-Kadec-Avdonin']}
  15. Proof of Lemma \ref{['lem:RB-basic-operations']}
  16. ...and 1 more sections

Key Result

Theorem 1

For nonsingular matrices $A,B \in \mathbb R^{d \times d}$, the following are equivalent: For $d\leq 7$, the statements above are also equivalent to:

Figures (4)

  • Figure 1: Both of the frequency sets in Theorem \ref{['thm:tensor-product-Kadec-Avdonin']} (left) and Theorem \ref{['thm:Bailey-Cor6-1']} (right) are perturbations of $\mathbb{Z}^2$. The left set is a tensor product while the right set is obtained by independent perturbation of points in $\mathbb{Z}^2$. The size of perturbation allowed in each coordinate direction is less than $1/4 = 0.25$ for the left set, and is less than $\frac{\ln(2)}{2 \pi } \approx 0.1103$ for the right set.
  • Figure 2: The lattice $\Lambda = H \mathbb{Z}^2$ and its dual lattice $\Lambda^* = H^{-T} \mathbb{Z}^d$
  • Figure 3: The lattice $\Lambda^*$ (black dots) and its rounded set $\mathcal{C} = \mathbf{r} (\Lambda^*)$ (blue crosses)
  • Figure 4: The set $S = H [0,1]^2$

Theorems & Definitions (32)

  • Theorem 1: Corollary 7 in LPW24-first-cubes
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 5: see Definition 3.6.1 and Theorem 3.6.6 in Ch16, or Definition 7.9 and Theorem 8.32 in He11
  • Proposition 6: Kadec's $\frac{1}{4}$-theorem Ka64, see also Theorem 14 on p. 36 in Yo01
  • Proposition 7: Avdonin's theorem of "$\frac{1}{4}$ in the mean" Av74, also see the notes on p. 178 in Yo01
  • Theorem 8: cf. Theorem 2.1 in SZ99-ACHA, Theorem 1.1 in SZ99-JMAA
  • Theorem 9: Corollary 6.1 in Ba10; the case $d=1$ was proved by Duffin and Eachus DE42, see also Remarks on p. 37 in Yo01
  • Remark 10
  • ...and 22 more