Hermite Expansion Framework

• The Hermite Expansion Framework is a unified analytic and algebraic approach that characterizes spaces of functions with nearly optimal Gaussian decay.
• It establishes a direct link between time-frequency localization and exponential decay of Hermite-series coefficients via the Bargmann transform and Phragmén–Lindelöf principles.
• The framework provides a scalable methodology for extending classical and generalized Pilipović spaces, offering robust tools for spectral localization and functional analysis.

The Hermite Expansion Framework provides a unified analytic, algebraic, and functional-analytic machinery for characterizing, approximating, and analyzing spaces of functions and distributions on ℝ exhibiting nearly optimal Gaussian decay in both time and frequency domains. The recent work of Neyt, Toft, and Vindas establishes a definitive expansion-theoretic characterization of such functions, directly linking their time-frequency localization to the exponential decay of Hermite-series coefficients, and placing the result within a Bargmann-transform and weighted Phragmén–Lindelöf principle context (Neyt et al., 2024).

1. Definition of Nearly Optimal Gaussian Spaces

Let 𝒮(ℝ) denote the Schwartz space and 𝒮′(ℝ) its dual, the tempered distributions. The focus is on the Fréchet space ℰ defined by the family of Banach seminorms

$$p_λ(f) := \sup_{x∈ℝ} |f(x)| e^{(½−λ)x²} + \sup_{ξ∈ℝ}\,|\widehat{f}(ξ)| e^{(½−λ)ξ²}, \qquad λ > 0,$$

where $\widehat{f}(\xi)$ is the unitary Fourier transform. One sets

$$ℰ := \bigcap_{λ>0} \left\{ f∈C^∞(ℝ) : p_λ(f)<∞ \right\}.$$

This is a nuclear Fréchet space. Functions $f∈ℰ$ satisfy, for every $λ>0$, Gaussian decay in both variables: $$|f(x)| ≤ C_λ e^{−(½−λ)x²},\quad |\widehat{f}(ξ)| ≤ C_λ e^{−(½−λ)ξ²},\quad ∀ x,ξ∈ℝ.$$ No nonzero $f$ can satisfy a stronger exponential—by Hardy’s uncertainty principle, this decay is "nearly optimal".

2. Hermite Expansion Theorem and Space Identification

The physicists’ Hermite functions $\{H_n(x)\}_{n=0}^∞$, normalized in $L^2(ℝ)$ by

$$H_n(x) = (2ⁿ n! \sqrt{π})^{-½} (−1)^n e^{x²}\frac{d^n}{dx^n} e^{-x²},$$

form a complete orthonormal system. Every $f∈L^2(ℝ)$ admits

$$f(x) = \sum_{n=0}^∞ c_n H_n(x), \qquad c_n = \langle f, H_n\rangle_{L^2}.$$

Theorem A (Neyt–Toft–Vindas):

$f∈ℰ$ (in fact, $f∈L^2∩ℰ$) if and only if for every $r > 0$,

$|c_n| \lesssim e^{−r n},\qquad n=0,1,2,…$

In this case, the Hermite expansion converges in both $\mathcal E$ and $L^2$.

Thus,

$\mathcal E = H_{0,\,½}(ℝ),$

where $H_{0,\,½}(ℝ)$ denotes the smallest Fourier-invariant proper Pilipović space characterized by rapid-exponential Hermite coefficient decay.

Weighted Generalization

For a nondecreasing weight function $\omega:[0,∞)→[0,∞)$ satisfying subadditivity, integrability, and convexity criteria ((α), (β), (δ)), one defines the Young conjugate $\varphi^*$ and scale spaces

$$H_{ω,r} = \left\{ f∈L^2(ℝ) :\, \sup_{n≥0} |⟨f,H_n⟩| n!^{−½} e^{r \varphi^*(r n)} < ∞ \right\},$$

$$H_{ω} := \lim_{r→∞} H_{ω,r},\qquad H_{0,ω} := \bigcap_{r>0} H_{ω,r}.$$

Theorem B (Refined Hermite expansion):

$f∈L^2(ℝ)$ satisfies

$$|f(x)| ≤ C e^{−x²/2 +λ ω(|x|)},\quad |\widehat{f}(ξ)| ≤ C e^{−ξ²/2 +λ ω(|ξ|)}$$

for some (resp. every) $λ>0$ if and only if

$|c_n| ≤ C' n!^{−½} \exp[ −r \varphi^*(r n) ]$

for some (resp. every) $r>0$.

Selecting particular weights $\omega$ recovers the entire family of proper Pilipović spaces $H_s(ℝ)$.

3. Analytic Tools: Bargmann Transform and Phragmén–Lindelöf Principles

The central analytic step is to apply the Bargmann transform: $$\mathcal B f(z) := π^{-¼} \int_{ℝ} f(t)\,e^{−(z²+2tz−t²)/2}\,dt,\quad z∈ℂ,$$ where $\mathcal B H_n(z)=z^n/\sqrt{n!}$, yielding

$\mathcal B f(z) = \sum_{n=0}^∞ c_n\,z^n/\sqrt{n!}.$

The connection between time-frequency decay and the analytic growth properties of $\mathcal B f$ on $\mathbb C$ is established via two pivotal lemmas:

• Time–frequency ⇒ Bargmann: Time-frequency Gaussian bounds imply growth bounds for $\mathcal B f$ along real and imaginary axes.
• Bargmann ⇒ Time–frequency: If the Bargmann transform is entire and satisfies certain subexponential bounds, $f$ and $\widehat f$ inherit (weighted) Gaussian decay.

A sharp weighted Phragmén–Lindelöf principle on sectors then propagates these edge-bounds to the whole plane, yielding control on all Taylor/Hermite coefficients via the Cauchy inequality: $|c_n| \leq C n!^{-½} \exp[ -r \varphi^*(rn) ].$

4. Relations to Fourier Characterizations of Pilipović Spaces

Earlier descriptions of Pilipović spaces $H_s$ ($0≤s≤½$) employed mixed fractional Fourier transforms and tailored Fourier decay conditions [JFA 284 (2023) 109724]. The Hermite expansion framework:

• Replaces the need for partial or fractional Fourier transforms with explicit coefficient bounds,
• Transparently equates exponential Hermite-decay with time-frequency Gaussian bounds,
• Produces optimal exponential constants via the Phragmén–Lindelöf argument,
• Extends beyond classical power-type weights to any subadditive–convex $\omega$.

In the extremal case $s=½$, the largest proper Pilipović space matches $\mathcal E$, the "extremal" Gaussian space.

The table below summarizes the relationship:

Space	Time-Frequency Bound	Hermite Coefficient Bound
$\mathcal E$ ($H_{0,1/2}$)	$e^{-(\frac12-\lambda)x^2}$ in $x$ and $\xi$	$|c_n| \lesssim e^{-r n}$
$H_{0,\omega}$	$e^{-x^2/2+\lambda\omega(x)}$	$|c_n| \lesssim n!^{-½}\exp(-r\varphi^*(rn))$

5. Structural and Functional Analytic Implications

The nuclear Fréchet structure of $\mathcal E$ derives from the system of seminorms indexed by $\lambda>0$, guaranteeing excellent topological and duality properties. The Hermite expansion provides a natural analytic basis, with temporal-frequency behavior and expansion coefficients tightly coupled.

Key consequences include:

• Sharpness: The equivalence between nearly-optimal decay and rapid-exponential Hermite coefficient falloff is both necessary and sufficient—the bounds are tight.
• Scalability: The method accommodates sub-Gaussian and broader weights $\omega$.
• Completeness of Framework: No functions with stronger simultaneous decay in $x$ and $\xi$ exist by Hardy's theorem.

6. Connections and Extensions

The Hermite expansion framework established by (Neyt et al., 2024) positions the Hermite basis as the canonical organizing principle for spaces with nearly-optimal time-frequency localization. The methods—especially via the Bargmann transform and analytic function growth principles—ensure both theoretical and practical robustness, subsuming and refining preceding Fourier-based analysis for Pilipović spaces.

This framework is foundational for further investigations into function spaces characterized by ultrarapid decay, spectral and phase-space localization, and their associated dual spaces of generalized functions and distributions.