Weierstrass Transform Overview

• Weierstrass transform is a classical integral operator that smooths functions via convolution with a Gaussian kernel.
• It plays a crucial role in harmonic analysis and probability theory, linking to the Ornstein–Uhlenbeck semigroup and Hermite polynomial expansions.
• Its analytic properties underpin the equivalence of Hermite and power ranks, and highlight sensitivity to constant shifts in function behavior.

The Weierstrass transform is a classical integral transform acting on functions defined on the real line, central to probability theory, harmonic analysis, and @@@@1@@@@. Its generalized formulation connects deeply with the theory of Hermite polynomials and the Ornstein–Uhlenbeck semigroup, providing a natural smoothing operation via convolution with a Gaussian kernel. This transform plays a key role in characterizing analytic properties of functions in L² spaces under Gaussian measures and underpins the equivalence of Hermite rank and power rank in the Gaussian case (Bai et al., 2016).

1. Formal Definition and Equivalent Representations

Let $f: \mathbb{R} \to \mathbb{R}$ be any locally integrable function of at most exponential growth. The generalized Weierstrass transform $W_t[f]$ with parameter $t > 0$ is defined by convolution against a centered Gaussian of variance $t$: $W_t[f](x) := \mathbb{E}[f(x+\sqrt{t} Z)]$ where $Z \sim N(0,1)$. In integral form,

$$W_t[f](x) = \int_{\mathbb{R}} f(y) \, \frac{1}{\sqrt{2\pi t}} \exp\left(-\frac{(y-x)^2}{2t}\right) dy.$$

For any continuity point $x$ of $f$, $W_t[f](x) \to f(x)$ as $t \to 0^+$. In $L^2(\gamma)$ (with $\gamma$ the $N(0,1)$ law), $W_t$ can be inverted via Hermite polynomial expansion. If

$f(x) = \sum_{k=0}^\infty a_k H_k(x),$

then

$$W_t[f](x) = \sum_{k=0}^\infty a_k e^{k t/2} H_k(x),$$

and the inversion is given formally by

$$a_k = e^{-k t/2} \langle W_t[f], H_k \rangle_{L^2(\gamma)}, \quad f(x) = \sum_{k=0}^\infty e^{-k t/2} \langle W_t[f], H_k \rangle H_k(x).$$

Here, $H_k$ denotes the $k$-th probabilists’ Hermite polynomial (Bai et al., 2016).

2. Fundamental Properties

The Weierstrass transform exhibits several fundamental operator-theoretic and analytic properties:

• Linearity: $$W_t[\alpha f + \beta g] = \alpha W_t[f] + \beta W_t[g]$$ for all $\alpha, \beta \in \mathbb{R}$.
• Semigroup Property: $W_s \circ W_t = W_{s+t}$ for $s, t \geq 0$, $W_0 = Id$.
• Smoothing: For any locally integrable $f$, $W_t[f]$ is $C^\infty$ in $x$. Differentiating under the integral,

$$\partial_x^m W_t[f](x) = t^{-m/2} \mathbb{E}[f(x+\sqrt{t} Z) H_m(Z)].$$

• Heat Equation Solution: $W_t$ solves the Cauchy problem $\partial_t u = \frac{1}{2}\partial_{xx}^2 u$, with $u(0,x) = f(x)$.
• Action on Monomials: For each $m \geq 0$,

$$W_t[x^m] = \sum_{j=0}^{\lfloor m/2 \rfloor} \frac{m!}{(m-2j)!\,j!} \left(\frac{t}{2}\right)^j x^{m-2j}.$$

• Action on Hermite Polynomials: Since $H_k$ is an eigen-basis for the Ornstein–Uhlenbeck generator $L = \frac{1}{2}(\partial^2 - x\partial)$, by Mehler’s formula under the parameterization $P_s = e^{sL}$, $P_s H_k = e^{-k s} H_k$. The relationship between the Weierstrass and OU semigroups is a time reparameterization (Bai et al., 2016).

3. Hermite Rank and Power Rank: Definitions and Equivalence

Given $Z \sim N(0,1)$ and any $f \in L^2(\gamma)$, the Hermite expansion

$$f(x) = \sum_{k=0}^\infty a_k H_k(x), \quad a_k = \mathbb{E}[f(Z) H_k(Z)] / k!$$

characterizes $f$ in the orthonormal Hermite basis.

• Hermite Rank: The smallest integer $m \geq 0$ for which $a_m \neq 0$.
• Power Rank: For $f \in C^\infty$ in a neighborhood of $0$, the smallest $\ell \geq 0$ such that $f^{(\ell)}(0) \neq 0$.

The generalized Weierstrass transform provides the analytic bridge between these concepts. The function $W_t[f](x)$ can be expanded both in Hermite polynomials and in $x$-Taylor series. The first nonzero coefficient in either expansion coincides, establishing the equivalence of Hermite and power ranks in the Gaussian case (Bai et al., 2016).

4. Detailed Equivalence Argument

For $t > 0$, define $\Phi(t) := \mathbb{E}[f(\sqrt{t} Z)]$. Expanding $f(\sqrt{t} Z)$ via Hermite series and employing orthogonality,

$$\Phi(t) = \sum_{k=0}^\infty a_k \mathbb{E}[H_k(Z)] t^{k/2}.$$

Noting that $\mathbb{E}[H_k(Z)] = 0$ for $k \geq 1$, only the constant term survives: $\Phi(t) = a_0$. To access higher Hermite coefficients, examine $\partial_x^\ell W_t[f](0)$: $$\partial_x^\ell W_t[f](0) = t^{-\ell/2} \mathbb{E}[f(\sqrt{t} Z) H_\ell(Z)].$$ Because $H_\ell(Z)$ is a degree-$\ell$ polynomial, for small $t$ the leading contribution arises from the $\ell$th derivative of $f$ at 0. Thus, the location of the first nonzero Hermite coefficient matches the order of the first nonzero derivative at the origin. Hence, $\mathrm{Hermite\ rank} = \mathrm{power\ rank}$ for such $f$ (Bai et al., 2016).

5. Instability of High Rank Under Constant Shifts

Suppose $f$ has Hermite rank at least $2$ (equivalently, $f(0) = f'(0) = 0$, but $f''(0) \neq 0$). For the shifted function $f_c(x) = f(x + c)$,

$$f_c(x) = f(c) + f'(c)x + \frac{1}{2}f''(c)x^2 + \cdots,$$

unless $c$ is simultaneously a root of $f(c) = 0$ and $f'(c) = 0$, the constant or linear coefficient of $f_c$ is nonzero. Consequently, the power rank (and hence Hermite rank) of $f_c$ drops to $0$ or $1$. Arbitrarily small shifts destroy Hermite rank $\geq 2$. Explicitly, for $f(x) = x^2 - 1$ (Hermite rank $2$), $$f_\varepsilon(x) = (x + \varepsilon)^2 - 1 = x^2 + 2\varepsilon x + (\varepsilon^2 - 1)$$, which has Hermite rank $1$ unless $\varepsilon = 0$ (Bai et al., 2016).

6. Illustrative Examples

Several explicit cases exemplify the Weierstrass transform and associated Hermite/power rank:

Function $f(x)$	$W_t[f](x)$	Hermite Rank = Power Rank
$x^m$	Polynomial (see monomial formula)	$m$
$e^{\lambda x}$	$e^{\lambda x + \frac{1}{2}\lambda^2 t}$	$0$
$x^3 + x$	Regularized; $H_1$ survives	$1$

In each case, the first nonzero coefficient in the Hermite expansion matches the first nonzero derivative in the Taylor expansion at the origin, establishing the structural identity at the center of the analysis (Bai et al., 2016).