Physlib.SpaceAndTime.Space.Norm.Regularized

Given a dimension $d$, a regularization parameter $\epsilon \in \mathbb{R}$, and an exponent $s \in \mathbb{R}$, this function maps an element $x$ of the $d$-dimensional space $\text{Space } d$ to the value $(\|x\|^2 + \epsilon^2)^{s/2}$, where $\|x\|$ denotes the Euclidean norm on $\text{Space } d$.

For any natural number $d$ and real numbers $\epsilon$ and $s$, the function $\text{normRegularizedPow}(d, \epsilon, s)$ on $\text{Space } d$ is given by the expression $x \mapsto (\|x\|^2 + \epsilon^2)^{s/2}$, where $\|x\|$ denotes the Euclidean norm on $\text{Space } d$.

For any natural number $d$, any element $x$ in the $d$-dimensional space $\text{Space } d$, and any non-zero real number $\epsilon$, the sum of the square of the norm of $x$ and the square of $\epsilon$ is strictly positive: $$\|x\|^2 + \epsilon^2 > 0$$ where $\|x\|$ denotes the Euclidean norm of $x$.

For any natural number $d$, any non-zero real number $\epsilon$, any real number $s$, and any element $x$ in the $d$-dimensional space $\text{Space } d$, the regularized norm power $(\|x\|^2 + \epsilon^2)^{s/2}$ is strictly positive: $$(\|x\|^2 + \epsilon^2)^{s/2} > 0$$ where $\|x\|$ denotes the Euclidean norm of $x$.

For any dimension $d \in \mathbb{N}$, any non-zero real number $\epsilon \in \mathbb{R}^\times$, and any real exponent $s \in \mathbb{R}$, the regularized norm power function $x \mapsto (\|x\|^2 + \epsilon^2)^{s/2}$ defined on the $d$-dimensional space $\text{Space } d$ has temperate growth, where $\|x\|$ denotes the Euclidean norm on $\text{Space } d$.

For any dimension $d \in \mathbb{N}$ and real numbers $\epsilon, s \in \mathbb{R}$, the regularized power of the norm function $x \mapsto (\|x\|^2 + \epsilon^2)^{s/2}$ from $\text{Space } d$ to $\mathbb{R}$ is measurable, where $\|x\|$ denotes the Euclidean norm on $\text{Space } d$ and the space is equipped with its Borel $\sigma$-algebra.