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When I can truncate a function space to a subspace, in a way that non-negative functions stay non-negative?

How I got here (a simple concrete example):

I was working with point-process intensity functions $\rho(x)$ on $x\in\mathbb R^n$. I wanted to find some "nice" convolutional filters $f(x)$ that:

  • Send all Fourier frequencies $\omega \in \mathcal \Omega$ above $\omega_0$ to identically zero, that is $\|\omega\|>\omega_0 \Rightarrow \{\hat f\cdot \hat \rho\}(\omega)=0$ (to not worry about high-frequency details).
  • Are themselves non-negative $f(x)\ge 0$ (to interpret $f*\rho$ as the PDF resulting from addition of improper random variables).
  • $f(x)$ and $\hat f(\omega)$ are "as nice as possible" in any interesting sense. Probably things like "unimodal", and "peak at zero" "as concentrated as possible near zero".

You can get some very nice low-pass $f(x)$ with compact support on $\|\omega\|\ge\omega_0$, achieve their maximum at $x=0$, are reasonably concentrated near $x=0$, and have $\nabla f(x)=0$ for $x=0$. For example, $f(x) \propto \operatorname{sinc}(\alpha x)^4$ is nice. I wasn't able to find any such $f(x)$ that decayed monotonically from $x=0$; Can such a function exist? I realized I didn't have the tools to reason about this.

My attempts to generalize

This seemed like a special case of something that may have relevant results in functional analysis. Here is my best-attempt at generalizing the problem:

  • Let $\rho \in \mathcal R:\mathcal X\mapsto \mathbb R_{\ge 0}$ be non-negative functions on $x \in \mathcal X$.
  • Let $f \in \mathcal F:\mathcal X\mapsto \mathbb R$ be scalar functions on $x \in \mathcal X$.
  • Let $\mathcal M$ a function space contining $\mathcal R$ and $\mathcal F$.
  • Let $U:\mathcal M\mapsto \mathcal M$ be operators that preserve non-negativity: $U\rho \subseteq \mathcal R$
  • Let $K:\mathcal M\mapsto \mathcal M$ be operators that send a non-trivial subspace of $\mathcal M$ to zero (have a nontrivial kernel).
  • Q: How can we find operators that are in both $U$ and $K$?

The definitions above should be considered "purely formal", "vague", or "confused, please help me fix it".

I'm interested in keywords, the names of mathematical objects, well-known results, topics and sub-fields that I can search for to learn more? What theorems, concepts, and results on this topic might I find empowering?

MRule
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