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., 6 May 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., 6 May 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.