Zak Transform of Hermite Functions

• Zak transform of Hermite functions is defined via a unitary, quasi‐periodic sum that connects to theta functions, enabling explicit Gabor frame analysis.
• It exhibits clear periodicity and symmetry with zero-sets determined by Hermite function parity, directly affecting frame properties in L²(ℝ).
• These structural insights reduce complex Gabor system investigations to standard lattice problems, guiding practical oversampling and frame construction.

The Zak transform of Hermite functions is a central tool in the mathematical analysis of Gabor systems, providing explicit series expansions and structural insights critical for the study of time–frequency localization and frame properties in $L^2(\mathbb{R})$. Hermite functions $h_n$ form a complete orthonormal set in $L^2(\mathbb{R})$ and their Zak transforms offer closed-form connections to theta functions and Gabor frame theory. This entry presents the definition, explicit formulas, periodicity, zero-sets, symmetries, and the frame-theoretic consequences of the Zak transform for Hermite functions, emphasizing its role in the characterization and construction of Gabor frames.

1. Definition and Normalization of the Zak Transform

For $f \in W_0(\mathbb{R})$, the Zak transform is given by

$$\mathcal{Z}f(x,\omega) = \sum_{k\in\mathbb{Z}} f(k - x)\, e^{2\pi i\,\omega k}, \qquad (x,\omega)\in\mathbb{R}^2.$$

On $L^2(\mathbb{R})$, this transform is unitary and plays a pivotal role in diagonalizing Gabor frame operators over integer lattices (Faulhuber, 12 Feb 2025, Faulhuber, 13 Feb 2025). The Zak transform admits quasi-periodicity relations: $$\mathcal{Z}f(x+1,\omega) = e^{-2\pi i\omega} \mathcal{Z}f(x,\omega), \quad \mathcal{Z}f(x,\omega+1) = \mathcal{Z}f(x,\omega).$$ For time–frequency shifts $\pi(\xi, \eta) = \mathcal{M}_\eta \mathcal{T}_\xi$, the Zak transform satisfies

$$\mathcal{Z}(\pi(\xi,\eta)f)(x,\omega) = e^{-2\pi i x \eta}\, \mathcal{Z}f(x+\xi,\omega+\eta).$$

These properties underpin its role in the study of lattice-generated Gabor systems.

2. Explicit Formulas for the Hermite Zak Transform

The $n$-th Hermite function is

$$h_n(t) = (-1)^n C_n\, e^{\pi t^2} \frac{d^n}{dt^n} e^{-2\pi t^2},$$

where the normalization is $C_n = 2^{1/4}/\sqrt{(2\pi)^n 2^n n!}$ (Faulhuber, 12 Feb 2025). Its Zak transform possesses the explicit expansions: $$\mathcal{Z}h_n(x,\omega) = C_n\, \sum_{k\in\mathbb{Z}} H_n(\sqrt{2\pi}(k-x))\, e^{-2\pi(k-x)^2}\, e^{2\pi i\,\omega k},$$ where $H_n$ is the physicists' Hermite polynomial, and equivalently, in terms of the Jacobi theta-3 function $$\vartheta_3(z;\tau) = \sum_{k\in\mathbb{Z}} e^{\pi i\tau k^2 + 2\pi i k z}$$,

$$\mathcal{Z}h_n(x,\omega) = \frac{C_n}{(2\pi i)^n\, e^{-\pi x^2}}\, \partial_z^n \vartheta_3(z; i)\big|_{z = \omega + i x}.$$

These expansions connect Hermite Zak transforms to classical special functions and facilitate analytic studies of their properties (Faulhuber, 12 Feb 2025).

3. Periodicity, Symmetry, and Zero-Sets

The Zak transform for Hermite functions exhibits quasi-periodicity in $(x,\omega)$ on $\mathbb{R}^2$, and its structural zeros are determined by the parity of $h_n$:

• If $n$ is even, $\mathcal{Z}h_n(\tfrac{1}{2},\tfrac{1}{2}) = 0$ is the unique parity zero on $[0,1)^2$.
• If $n$ is odd, zeros appear at $(0,0), (\tfrac{1}{2},0), (0,\tfrac{1}{2})$ in $[0,1)^2$ (Faulhuber, 13 Feb 2025).
• For $n \equiv 2 \mod 4$, $$\mathcal{Z}h_{4\ell+2}(0,0) = \mathcal{Z}h_{4\ell+2}(\tfrac{1}{2},\tfrac{1}{2}) = 0$$.
• If $n \equiv 1,3 \mod 4$, then also $\mathcal{Z}h_n(\tfrac{1}{2},\tfrac{1}{2})=0$.
• For $n \equiv 0 \mod 4$, it may happen that $\mathcal{Z}h_n(0,0) \neq 0$ but always $\mathcal{Z}h_n(\tfrac{1}{2},\tfrac{1}{2})=0$ (Faulhuber, 12 Feb 2025).

The origin of these zeros is the interplay between Hermite function parity, Poisson summation, and Zak transform properties [Jan88] [Gro01]. For $n\geq 3$, additional non-parity zeros may arise within $[0,1)^2$ (Faulhuber, 13 Feb 2025).

Zeros Table

$n$ Parity	Parity Zeros in $[0,1)^2$	Further Structure
Even	$(\tfrac{1}{2},\tfrac{1}{2})$	$n\equiv 2\,(\text{mod}\,4)$: extra zero at $(0,0)$
Odd	$(0,0),\; (\tfrac{1}{2},0),\; (0,\tfrac{1}{2})$	More zeros possible for $n\geq 3$

4. Unitary Equivalences and Reduction to Standard Lattice Problems

Metaplectic–symplectic pairs $(\mathcal{U}, U)$, including the dilation $(\mathcal{D}_a, D_a)$, chirp $(\mathcal{V}_q, V_q)$, and fractional Fourier transform $(\mathcal{F}_r, R_r)$, generate unitary equivalences of Gabor systems: $$\mathcal{U}\, \pi(z)\, \mathcal{U}^{-1} = c_U(z)\, \pi(Uz), \quad \mathcal{G}(g, \Lambda) \simeq \mathcal{G}(\mathcal{U}g, U\Lambda).$$ This framework enables reduction of Gabor analysis for Hermite functions on arbitrary lattices to problems on the "standard" Zak strip $[0,1)^2$ by suitable choice of window and lattice transformation (Faulhuber, 12 Feb 2025). For $n=0,2,4,\dots$, the fractional Fourier transform relates higher Hermite functions to the Gaussian by a change of window class, thus simplifying certain analyses to the $h_0$ case.

5. Implications for Gabor Frames and Oversampling

The frame property for Gabor systems generated by Hermite functions is governed by the behavior of their Zak transforms. The fundamental frame criterion for a multi-window Gabor system over $\mathbb{Z}^2$ asserts: $$\mathcal{G}_M(\{g_m\}, \mathbb{Z}^2)\ \text{is a frame} \iff 0 < A \leq \sum_{m=1}^M |\mathcal{Z} g_m(x, \omega)|^2 \leq B < \infty \text{ on } [0,1)^2.$$ Zeros of $\mathcal{Z}h_n$ preclude the lower frame bound and thus rule out the frame property for certain configurations. For example:

• For $n \equiv 2 \mod 4$, the singly oversampled system $\mathcal{G}(h_{4\ell+2}, (1/\sqrt{2})\mathbb{Z}^2)$ is not a frame [Lem16; (Faulhuber, 12 Feb 2025)].
• With sufficiently chosen shifts, multi-window constructions can overcome the presence of Zak zeros and restore the frame property.
• Sampling densities $\rho > n+1$ do not always guarantee the frame property: for every $n \geq 1$, there exist periodic index sets $\Gamma$ with density $M>n+1$ such that $G(h_n, \Gamma)$ fails the lower-frame inequality (Faulhuber, 13 Feb 2025). This refines the classical density criterion [Groechenig–Lyubarskii_Hermite_2007].

6. Examples and Frame Failure by Zak Zeros

Concrete examples demonstrate the nexus between Zak zeros and Gabor frame failure:

• $n=0$ (Gaussian): only parity zero at $(\frac{1}{2},\frac{1}{2})$, reproducing the classical non-frame result for $G(h_0, \mathbb{Z}^2 + (\frac{1}{2},\frac{1}{2}))$.
• $n=1$: three parity zeros result in non-frame for $$G(h_1, \mathbb{Z}^2 \cup (\mathbb{Z}^2 + (\frac{1}{2},0)) \cup (\mathbb{Z}^2 + (0,\frac{1}{2})))$$ at density $3>2$.
• $n=2$: single parity zero, aligns with the lattice case.
• $n=3$: multiple zeros facilitate the construction of non-frame Gabor systems even for densities exceeding $n+1$ (Faulhuber, 13 Feb 2025).

The link between the number of Zak zeros within $[0,1)^2$ and the possibility of constructing high-density, non-frame Gabor systems underlines the structural importance of the Zak transform zeros.

7. Significance, Extensions, and Open Problems

The explicit and structural results for the Zak transform of Hermite functions provide a rigorous framework for understanding Gabor frame properties, oversampling phenomena, and the geometry of time–frequency representations. The unitary equivalence approach allows transfer of results between lattices and window classes, while the explicit theta function connection facilitates analytic and computational work. Open directions include the full classification of Zak zero sets for general $n$, the extension to other function systems, and further refinements of density criteria. Comprehensive treatments and further proofs are contained in (Faulhuber, 12 Feb 2025) and (Faulhuber, 13 Feb 2025).