- Given a polynomial of the form $\sum a_n x_n^n$, are we able to guarantee the existance of a second polynomial $\sum b_n x_n^n$ such that $\sum b_n x_n^n = |\sum a_n x_n^n|$? This is of interest when considering using squared polynomials to force a function to have positive values. We may want to talk about the log of a squared polynomial, and for this we definitely do not want negative values.
- Does there exist a set of basis functions $h_n(x)$ such that when given a linear combination $\sum c_n h_n(x)$ we precisely know the minimum value?
- Taking a pdf and multiplying it by $e^{x^2}$ and then taking the square root, we have something that our basis functions from yesterday can interpolate. What does this modified pdf on its own tell us? This is a nice way to map PDFs to general functions, and a bijection to $C^{\infty}$ when the pdf is in $C^{\infty}$
Questions on top of mind
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Ok, here's the idea. Suppose I had a function $$f:X \to Y$$ and distance metrics $d_X:X \to \mathbb{R}$ and $d_Y:Y \to \mathbb{R}$. If ...
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Here's some code to generate those basis functions from yesterday. As you might imagine, you need to increase the decay rate as you ask ... -
Using the family of functions we defined a couple days ago of the form $f_n\left(x\right) = \sum^n_{k=0} c_{k,n} x^k e^{-\frac{x^2}{2}}$ we ...
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