Fourier–Bohr Transformation

• Fourier–Bohr transformation is a generalization of the Fourier series to almost periodic functions, measures, and point sets on various topological groups.
• It computes Fourier–Bohr coefficients using volume-averaged exponential sums or ergodic means, ensuring uniform convergence and precise spectral decomposition.
• This framework underpins diffraction theory and quasicrystal analysis by providing explicit formulas for Bragg peak intensities and pure-point spectra.

The Fourier–Bohr transformation generalizes the classical Fourier series to encompass almost periodic functions, measures, and point sets in various topological group settings, extending its domain to include Euclidean spaces, locally compact Abelian groups, solenoids, and model sets associated with quasicrystals. At its core, this transformation assigns to each admissible function or measure a countable (sometimes sparse) collection of coefficients—known as Fourier–Bohr coefficients—derived as volume-averaged exponential sums or ergodic means along group orbits. These coefficients encode the pure-point or singular spectral components and are fundamental in diffraction theory, the paper of quasicrystals, and the analysis of almost periodic phenomena. The rigorous framework supporting their existence, convergence, precise value, and structural consequences is provided by results in harmonic analysis, the theory of almost periodic functions, and cut-and-project schemes.

1. Frameworks and Definitions of Fourier–Bohr Coefficients

For a translation-bounded Radon measure $\mu$ on a second-countable locally compact Abelian group $G$, the Fourier–Bohr coefficient at a character $\chi \in \widehat G$ is defined as

$$c_\chi(\mu) := \widehat\mu(\{\chi\}) = \lim_{n\to\infty} \frac{1}{\theta_G(A_n)} \int_{A_n} \overline{\chi(x)}\, d\mu(x)$$

for any van Hove sequence $(A_n)$. For Dirac combs of regular model sets $A \subset \mathbb{R}^d$,

$$a(k) = \lim_{R \to \infty} \frac{1}{(2R)^d} \sum_{x \in A \cap B_R} e^{-2\pi i\,k \cdot x}$$

where $B_R$ is a cube of side $2R$ in $G$.

On solenoids $$S \cong (\mathbb{R} \times \widehat{\mathbb{Z}})/\mathbb{Z}$$, the Fourier–Bohr transform acts on continuous functions via

$$\widehat{\Phi}(\lambda, \rho) = M\Big( \Phi(x, t) \,\overline{\chi_{\lambda, \rho}(x, t)} \Big)$$

where $M(\cdot)$ is the invariant mean, and $\chi_{\lambda, \rho}$ ranges over the character group $\mathbb{R} \times (\mathbb{Q}/\mathbb{Z})$.

2. Main Convergence Theorems and Structural Results

For regular Euclidean model sets $$(G = \mathbb{R}^d, H = \mathbb{R}^{k-d}, L \subset G \times H)$$ with topologically regular window $W \subset H$ (i.e., $W = \overline{\operatorname{int}(W)}$, $\partial W$ has measure zero), the main theorem asserts:

• For every $k \in G$, the limit

$$a(k) = \lim_{R\to\infty} \frac{1}{(2R)^d} \sum_{x \in A \cap B_R} e^{-2\pi i\,k \cdot x}$$

exists and converges uniformly in $k$.

• The explicit value is

$$a(k) = \begin{cases} \operatorname{dens}(L)\, \widehat{1_W}(-k^*) & k \in L^\diamond \ 0 & k \notin L^\diamond \end{cases}$$

where $L^\diamond = T(L^*) \subset G$ is the Fourier module, $k^*$ is the star image in $H$.

• The diffraction measure is

$$\widehat{\gamma}_A = \sum_{k \in L^\diamond} |a(k)|^2\, \delta_k$$

This framework establishes the pure-point nature of the spectrum, central for quasicrystal diffraction (Baake et al., 2023).

In LCAGs, Fourier–Bohr coefficients exist for translation-bounded, Fourier-transformable measures if both support and FBS are locally finite, and measures supported inside Meyer sets exhibit additional crystallographic structure: their supports are finite unions of lattice cosets, and their transform is supported on the dual lattice $\Lambda^\perp$ plus finite translates. Positive-definite, doubly-sparse measures admit Poisson summation type formulae for their autocorrelation and diffraction (Baake et al., 2019).

3. Poisson Summation, Cut-and-Project Schemes, and Model Set Theory

The elementary proof for convergence of Fourier–Bohr coefficients in regular model sets employs:

• $\varepsilon$-mollifications of cube and window indicator functions.
• The Poisson summation formula on the lattice $L$:

$$\sum_{x \in \Lambda} F(x + y) = \operatorname{dens}(\Lambda) \sum_{\xi \in \Lambda^*} \widehat{F}(\xi) e^{2\pi i\,\xi \cdot y}$$

• Isolation of the principal term at zero frequency to obtain the main contribution; all other dual terms decay uniformly by exponential sum estimates.

Uniform distribution properties in model sets ensure density formulas and control for boundary terms. The density of a model set $A = X(W)$ is $\operatorname{dens}(L) \cdot \operatorname{vol}(W)$, with uniform convergence in volume for windows satisfying regularity (Baake et al., 2023).

For measures of toral type, associated with Bohr almost periodic sets, the decomposition hinges on a group compactification $\psi : \mathbb{R}^n \rightarrow \mathbb{T}^m$, where the Fourier–Bohr transform corresponds to a pure-point Fourier series on $\mathbb{T}^m$ pulled back via the dual map $\widehat{\psi}$ (Lawton, 2021).

4. Structure of Sparse and Doubly Sparse Measures

A measure is doubly sparse if both the support of the measure and its Fourier–Bohr spectrum are pure point and sparse (upper density finite). Under appropriate conditions (e.g., Meyer set support), both the support and FBS reduce to finite unions of lattice translates. The general representation is:

$$\mu = \sum_{i=1}^N \sum_{x \in \Lambda + T_i} P_i(x) \, \delta_x, \quad \widehat{\mu} = \sum_{j=1}^M \sum_{\chi \in \Lambda^\perp + F_j'} Q_j(\chi) \, \delta_\chi$$

where $P_i$, $Q_j$ are trigonometric polynomials, $F$, $F'$ finite sets, and $\Lambda$ is a lattice in $G$ (Baake et al., 2019).

For positive-definite measures supported on model sets, their autocorrelation and diffraction are given by weighted model combs and their Fourier transforms, encapsulating the Poisson summation formula as a particular case.

5. Fourier–Bohr Theory on Solenoids and Toral Compactifications

On one-dimensional solenoids $S$, continuous invariant functions admit an extension of the Bohr–Fourier theory, with coefficients computed via invariant means over both $\mathbb{R}$-leaves and the transversal $\widehat{\mathbb{Z}}$. The transformation identifies nonzero coefficients only on the diagonal $(\lambda, \rho)$ with $\rho = [\lambda] \in \mathbb{Q}/\mathbb{Z}$. The Bohr–Fourier series converges uniformly, and Parseval and uniqueness relations hold verbatim as in the classical case. The solenoidal setting enforces rational frequency pairing, distinguishing it from the classical real line analysis (Cruz-López et al., 2019).

On toral compactifications, measures of finite-rank toral type exhibit pure-point spectra. Associated Borel measures on components of $\mathbb{T}^m$ encode density and homotopy subgroup structure, which for $n = 1$ recovers the Olevskiĭ–Ulanovskiĭ characterization of one-dimensional Fourier quasicrystals.

6. Applications: Diffraction, Quasicrystals, and Almost Periodic Structures

The Fourier–Bohr transformation rigorously underpins the pure-point diffraction spectrum of model sets and Meyer sets, yielding explicit Bragg peak locations and intensities:

• Fibonacci chains: Bragg peaks correspond to $k = m + n/\tau$ for $(m, n) \in \mathbb{Z}^2$ and irrational $\tau$, with amplitudes given by sinc-ratio factors from the window's Fourier transform.
• Penrose tilings and higher-dimensional quasicrystals: Peak locations on arithmetic modules, intensities by the Fourier transform of the prototile window.
• Sparse pure-point measures: Their structure tightly constrains possible Fourier–Bohr supports; in Euclidean cut-and-project settings, only lattices admit locally finite, non-periodic doubly sparse transforms (negative answer to Meyer’s question) (Baake et al., 2019).

The Fourier–Bohr coefficients also determine the autocorrelation and diffraction measures: for Dirac combs of regular model sets, $\widehat{\gamma}_A$ is a purely atomic measure with intensities $|a(k)|^2$, corresponding to physical observables in diffraction experiments.

7. Comparison with Classical and Generalized Settings

Aspect	Classical ($\mathbb{R}$)	Model Set / Solenoid / Toral Type
Domain	$\mathbb{R}$	$$(\mathbb{R}^d, G),\; S = (\mathbb{R} \times \widehat{\mathbb{Z}})/\mathbb{Z}$$, $\mathbb{T}^m$
Characters	$e^{i\lambda x}$, $\lambda \in \mathbb{R}$	$(\lambda, \rho)$, diagonal $\rho=[\lambda]$; $e^{2\pi i k \cdot z}$, $k\in\mathbb{Z}^m$
Mean/limit	$\lim_{T\to\infty}\frac{1}{T}\int_0^T f(x) dx$	$\lim_{R\to\infty}$, van Hove or ergodic averages over cube/compact set
Coefficients	$a(\lambda) = M(f e^{-i\lambda x})$	$a(k)$ via volume-averaged sums; $\widehat{\Phi}(\lambda,\rho)$ via invariant mean; $\widehat{\kappa}(k)$ by group duality
Series expansion	$f(x) = \sum a(\lambda) e^{i\lambda x}$	$$\Phi(x,t) = \sum a(\lambda) e^{i\lambda x} \chi_{[\lambda]}(t)$$; $$\mu = \sum \widehat{\kappa}(k)\, \delta_{\widehat{\psi}(k)}$$
Convergence	Uniform	Uniform (model sets, solenoids, toral type)
Uniqueness	Yes	Yes

A plausible implication is that the Fourier–Bohr transformation, in all of these settings, provides a powerful analytic and algebraic tool for classifying and explicitly computing the pure-point spectral content of almost periodic structures—from quasicrystals and Meyer sets, to solenoidal and toral atomic measures.

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References: (Baake et al., 2023, Baake et al., 2019, Lawton, 2021, Cruz-López et al., 2019)