I would like to define the relationship between two functions: $f(y,x)$ and $g(y,x)$ - i.e. $g = Q(f) := f(y,x) \mapsto g(y,x) = (\mathbb{R}^2\mapsto \mathbb{R}) \mapsto (\mathbb{R}^2\mapsto \mathbb{R})$.

The motivation is this:

Given that $x\in\mathcal{X}$, consider two independent subsets: $\mathcal{X}_1 \subset \mathcal{X}$, $\mathcal{X}_2 \subset \mathcal{X}$.

We have observed $f(y,\mathcal{X}_1)$ and $g(y,\mathcal{X}_1)$, and also $f(y,\mathcal{X}_2)$, but not $g(y,\mathcal{X}_2)$. So, we would like to infer $g(y,\mathcal{X}_2)$ from $f(y,\mathcal{X}_2)$. Assume that the spans of $\mathcal{X}_1$ and $\mathcal{X}_2$ do not overlap at all (so it is not simply interpolation).

What is this type of mapping called, and where can I find more information?

  • 1
    $\begingroup$ $Q$ would usually be called an operator. Do you the exact details of $Q$, or are you trying to infer it as well? $\endgroup$ – Nick Alger Dec 17 '16 at 16:34
  • $\begingroup$ Thanks. I do not know the details of $Q$; this is what I'm trying to estimate from $f(y,\mathcal{X}_1)$ and $g(y,\mathcal{X}_1)$. Are there common parameterizations of operators like this? I expect to need more complexity than a linear transformation. $\endgroup$ – Jesse Knight Dec 17 '16 at 16:45

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