What if $H$ is a I've specified in 2. Your formula I don't think it works in that case.

What is $h$? Also how did you get the formula. I don't know if it does. I think the analogue I am thninking of this problem is when you want to find a matrix $X$ such that $\left\lVert XA - B \right\rVert_F^2$ is minimal.

I think what I am trying to work out is a formula or a procedure to construct a minimizer $H$.

I more meant "quadratic energies" not "quadratic forms". So something like $E(\alpha f) = \left| \alpha \right| ^2 E(f)$. Also in $\Delta f = 0$ I more meant the operator $\Delta$. I also think I meant if there's a general theorem between energies and selfadjoint operators (not necessarily quadratic).

Hi, thank you for the answer. I am assuming I need to assume separable Hilbert space for this theory, correct? I cannot see much theory generalized to Banach Spaces in general.

How is $f_{c;p}$ positive definite iff $p \leq 2$. Wouldn't this hold also for $p = 4$? (any even number).

I think I understand the idea. Which is a similar intuition as mine. What I was wondering however if there's some what to measure this ill conditioning (conditioning number could be one of it).

I am digesting the answer. Can you clarify what you mean by "characteristic function of normal distribution". Also what do you mean by "borderline positive definite", how are you assessing the borderline?

I don't think I fully got it. Please let me give some thoughts.

I gave it more thought and I wrote an update. Would be nice if someone checked I did not write rubbish.