Consider an approximation of unity $(\phi_\epsilon)_\epsilon $ defined such that :
$\forall \epsilon > 0, \quad \int_{\mathbb{R}^n} |\phi_{\epsilon}(x)| dx = 1$
$\forall R>0, \quad \lim_{\epsilon \rightarrow 0} \int_{||x || > R} |\phi_\epsilon(x)| dx = 0$
$ \forall f \in L^2(\mathbb{R}^n), \quad \lim_{\epsilon \rightarrow 0} || f - \phi_\epsilon * f ||_{L^2(\mathbb{R}^n)} = 0.$
In the sequel, we assume that this family is constructed using a single function $\phi: \mathbb{R}^n \rightarrow \mathbb{R}$ via the definition: \begin{equation} \phi_\epsilon = \epsilon^{-n}\phi(\cdot/\epsilon), \quad \quad (*) \end{equation} where $\phi$ is integrable with $L^1$ integral equal to $1$.
Let $f$ be a function in $L^2(\mathbb{R}^n)$, and $\Omega \subset \mathbb{R}^n$, define the mapping \begin{equation} \begin{array}{lcll} \Psi_{f,\Omega} \colon & \mathbb{R}^+ & \longrightarrow & \mathbb{R}^+ \\ & \epsilon & \longmapsto & ||f - \phi_{\epsilon}*f||_{L^2(\Omega)}^2 \end{array} \end{equation}
Aim: How to ensure the monotonicity of the mapping $\Psi_{f,\Omega}$ (at least in a neighbourhood of 0)? That is, what are the sufficient conditions on the function $\phi$ to guarantee that $$ \forall \epsilon_1 < \epsilon_2 \ll 1, \quad \Psi_{f,\Omega}(\epsilon_1) \leq \Psi_{f,\Omega}(\epsilon_2). $$
For example, if $\Omega = \mathbb{R}^n$, I can prove the monotonicity of the function $\Psi_{f,\mathbb{R}^n}$ by imposing the following conditions on the Fourier transform $\hat{\phi}$ of the function $\phi$:
- The function $\hat{\phi}$ is bounded on $\mathbb{R}^n$.
- $\hat{\phi}(\xi) < 1 $ for $\||\xi|| \ll 1$
- The function $\hat{\phi}$ is differentiable and for all $\xi \in \mathbb{R}^n$, $\xi^\top \hat{\phi}'(\xi) \leq 0.$
Indeed, by Parseval's Theorem $$ \Psi_{f,\mathbb{R}^n}(\epsilon) = ||\mathcal{F}\left(f - \phi_{\epsilon}*f \right)||_{L^2(\mathbb{R}^n)}^2 = ||\hat{f} - \hat{\phi_\epsilon} \hat{f}||_{L^2(\mathbb{R}^n)}^2 = || (1- \hat{\phi_\epsilon}) \hat{f}||_{L^2(\mathbb{R}^n)}^2 $$ By (*), $ \hat{\phi_\epsilon} = \hat{\phi}(\epsilon \cdot)$, then $$ \Psi_{f,\mathbb{R}^n}(\epsilon) = || (1- \hat{\phi}(\epsilon \cdot) \hat{f}(\cdot)||_{L^2(\mathbb{R}^n)}^2 $$ Now, since the function $\hat{\phi}$ is bounded and differentiable, the function $\Psi_{f,\mathbb{R}^n}$ is also differentiable and $$ \Psi_{f,\mathbb{R}^n}'(\epsilon) = -2 \int_{\mathbb{R}^n} \xi^\top \hat{\phi}'(\beta \xi)\, (1-\hat{\phi}(\beta \xi) )\,|\hat{f}(\xi)|^2 d\xi $$ Using the second and third condition on the function $\hat{\phi}$, we deduce that $\Psi_{f,\mathbb{R}^n}'(\epsilon) \geq 0$ for small values of $\epsilon$. Whence the monotonicity of the mapping $\Psi_{f,\mathbb{R}^n}$ in a neighborhood of $0$.
Recall that there are functions $\phi$ such that all the conditions stated above are fulfilled. An example is the multivariate Gaussian function \begin{equation} \label{exple phi Multivariate gaussian} \phi(x) = \frac{1}{\sqrt{\det(2 \pi \Sigma)}} \exp\left(-\frac{1}{2} x^\top \Sigma^{-1} x \right), \end{equation} with a diagonal covariance matrix $\Sigma = diag(\sigma_1,...,\sigma_n)$.
Main Problem: How can we ensure monotonicity of $\Psi_{f,\Omega}$ for a subset $\Omega$ of $\mathbb{R}^n$.
Please can someone help me ??