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I am interested in an explicit example of smooth partition of unity and I would greatly appreciate if someone could provide me an example for it. Let $[a, b]$ be an interval. I cover it with $U_i = (x_i - d_i, x_i + d_i)$ with $i=1,2,3$. And I would like to see a smooth partition of unity with these three sets. Any comments/suggestions would be appreciated. Thank you very much.

Johnny T.
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    are you not happy with wikipedia's picture of a partition of unity for the circle? https://en.wikipedia.org/wiki/Partition_of_unity – Calvin Khor Jan 02 '18 at 15:36
  • @CalvinKhor I would really like to see the functions defined! – Johnny T. Jan 02 '18 at 15:45
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    You can assume that these intervals $U_i$ don't overlap too much in the sense that $U_i$ is not a subset of $\cup_{j\neq i} U_j$. Otherwise you can eliminate one of them by taking constant $0$ function as a partition. In that case you can rearrange them so that we have $U_1, U_2, U_3, U_4,\ldots$ and only consecutive intervals intersect nonempty. With that you can build your partitions from smooth transitions from $0$ to $1$ (https://math.stackexchange.com/questions/328868/how-to-build-a-smooth-transition-function-explicitly) by piecewise glueing. – freakish Jan 02 '18 at 16:09

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Thrown together in Geogebra. You don't want to touch the b slider, sorry. Parameter a is how steep the bump function needs to be, so that the stuff happening in $U_1$ doesn't interact with the stuff in $U_3$. Parameter c is the size of $U_1\setminus \overline{U_2}$ which I took to also be the size of $U_3\setminus \overline{U_2}$. Allow me to leave what exactly $U_i$ are in this picture as an exercise...

g is your usual bump function that is 1 inside $U_1 \setminus \overline{U_2}$ and quickly drops off to 0 by the time it hits $\partial U_1$.

h is similarly made (actually I was lazy and translated a copy of g). Once $f_1=$g and $f_3=$h were created, $f_2=$p$=1-g-h$ is easy.

The dashed green line verifies that their sum is 1.

enter image description here

I've done nothing more than roughly follow the construction as laid out in text books, e.g. Lee's Manifolds.

Calvin Khor
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