Is there a family of mathematical curves that looks like the following?
Y1 = f1(X), Y2 = f2(X), …
Y(infinity) = 0, Y(-infinity)=0
-1 <= Y <= 1, or -1 < Y < 1, or combinations thereof
The first member of the family is roughly bell shaped, like a gaussian density function, and symmetrical and the peak is at (0,1)
The second has a zero at (0,0) and occupies the (-,-) and (+,+) quadrants and is kind of “backwards Z rotated sideways” shaped
The third has two zeros and is kind of M shaped
The nth has n-1 zeros
All members are alternately left-right symmetric or have 180 degree rotational symmetry around the origin
Note, maybe there’s a trivial zeroeth member that is a flat line Y=0 or maybe there’s a reason to say “zeroeth” where I said “first” or something.
I am asking because I think such a family would be very useful for modeling data that are interesting in the neighborhood of 0 but which should have tiny values far from zero. I think for example it’d be nifty to model such data as a linear combination of the first n members of this family.