- 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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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 ... -
A couple days ago I encountered a really neat solution to creating a basis using a Cholesky decomposition. Particularly, when we have a set...
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We hold the following truths to be self-evident (for real-matrices): $$\displaystyle A = U \Sigma V^T$$ $$\displaystyle A^+=V \Sigma^{-1} U...
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