PhyslibAlpha.ProbabilisticTheory.HilbertSpace.Unbounded.Existence.IteratedKernelGrowth

For any natural number $n \in \mathbb{N}$, there exists a non-negative constant $C \ge 0$ such that for every $y \in \mathbb{R}$, the $n$-th Hermite polynomial $H_n$ satisfies the growth bound: $$|H_n(y)| \le C(1 + |y|)^n$$ where $H_n(y)$ is the value of the $n$-th Hermite polynomial at $y$. According to the definition, a suitable choice for $C$ is the sum of the absolute values of the coefficients of $H_n$.

For any natural number $n \in \mathbb{N}$ and any positive real number $\epsilon > 0$, there exists a non-negative constant $D \ge 0$ such that for all $x \in \mathbb{R}$ satisfying $x^2 \le 1$ and for all $u \in \mathbb{R}$, the absolute value of the $(n+1)$-th derivative of the Gaussian kernel $g_\epsilon$ evaluated at $u - x$ is bounded by $$\left| \frac{d^{n+1}}{dt^{n+1}} g_\epsilon(t) \Big|_{t=u-x} \right| \le D (1 + |u|)^{n+1} \exp\left(-\frac{u^2}{2\epsilon}\right),$$ where the Gaussian kernel is defined as $g_\epsilon(t) = \frac{1}{\sqrt{\pi \epsilon}} \exp\left(-\frac{t^2}{\epsilon}\right)$. This bound is uniform for $x \in [-1, 1]$.

For any natural number $n \in \mathbb{N}$ and any positive real number $c > 0$, the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(u) = |u|^n \exp(-c u^2)$ is integrable.

For any natural number $n$ and any positive real number $c > 0$, the function $u \mapsto (1 + |u|)^n e^{-c u^2}$ is integrable on $\mathbb{R}$.

For any real number $\epsilon > 0$, the Gaussian kernel $g_\epsilon: \mathbb{R} \to \mathbb{R}$ defined by $g_\epsilon(t) = \frac{1}{\sqrt{\pi \epsilon}} \exp\left(-\frac{t^2}{\epsilon}\right)$ is infinitely differentiable (i.e., it is of class $C^\infty$).

For any natural number $n$ and any positive real number $\epsilon > 0$, the $n$-th iterated derivative of the Gaussian kernel $g_\epsilon(t) = \frac{1}{\sqrt{\pi \epsilon}} \exp\left(-\frac{t^2}{\epsilon}\right)$ is continuous on $\mathbb{R}$.

Let $g_\epsilon(t) = \frac{1}{\sqrt{\pi \epsilon}} \exp\left(-\frac{t^2}{\epsilon}\right)$ be the Gaussian kernel (or heat kernel). For any natural number $n$, any positive real number $\epsilon > 0$, and any $t \in \mathbb{R}$, the $n$-th iterated derivative of $g_\epsilon$, denoted $g_\epsilon^{(n)}$, is differentiable at the point $t$, and its derivative is equal to the $(n + 1)$-th iterated derivative $g_\epsilon^{(n+1)}(t)$.

Let $H$ be a complex Hilbert space. Suppose $U: \mathbb{R} \to \mathcal{B}(H)$ is a map such that for every $t \in \mathbb{R}$, the operator $U_t$ is unitary, and for every vector $\psi \in H$, the orbit $t \mapsto U_t \psi$ is continuous. Let $g_\epsilon(t) = \frac{1}{\sqrt{\pi \epsilon}} \exp\left(-\frac{t^2}{\epsilon}\right)$ be the Gaussian (heat) kernel for a given $\epsilon > 0$. Then for any $n \in \mathbb{N}$, the $n$-th derivative of the Gaussian kernel weighted by the unitary orbit, $t \mapsto g_\epsilon^{(n)}(t) U_t \psi$, is integrable.

Let $H$ be a complex Hilbert space. Let $U: \mathbb{R} \to \mathcal{L}(H)$ be a strongly continuous one-parameter group of unitary operators, satisfying $U_{s+t} = U_s U_t$ for all $s, t \in \mathbb{R}$. For $\epsilon > 0$, let $g_\epsilon(t) = \frac{1}{\sqrt{\pi \epsilon}} \exp(-t^2/\epsilon)$ be the Gaussian kernel and let $g_\epsilon^{(n)}$ denote its $n$-th derivative. For any vector $\psi \in H$, let $$\phi_{n, \epsilon} = \int_{-\infty}^{\infty} g_\epsilon^{(n)}(t) U_t \psi \, dt$$ be the Gårding vector of $\psi$ with respect to the $n$-th derivative of the Gaussian kernel. Then the orbit map $s \mapsto U_s \phi_{n, \epsilon}$ is differentiable at $s = 0$, and its derivative is given by $$- \int_{-\infty}^{\infty} g_\epsilon^{(n+1)}(u) U_u \psi \, du$$