 # Is there a family of mathematical curves that looks like this?

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

Most importantly,
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.

Wait, maybe I got my answer already from a friend I met in the hall. Separate out the jobs of creating zeros and creating tiny limits, by multiplying X^n by a function that is zero in the limits, like 1/(e^X + e^-X). Gonna go try some of these…

Seems so obvious. Why didn’t I think of that in the first place??

The requirements sound similar to eigenstate wavefunctions of the quantum harmonic oscillator. (Illustration). Which are basically the Hermite Polynomials multiplied by exp(-x^2) modulo some scalings.

our descriptions aren’t at all clear. It’s not even clear if your functions have one or two arguments as that seems to vary depending on which property you’re talking about. Nevertheless, I think maybe your first function might be fit by this

f(x) = k exp(-c|x|[sup]a[/sup])

where k is chosen to make sure this is a proper distribution with total probability one.

If you mean instead its a function of two variables and the function takes on it’s maximum value when x = 0 and y = 1 then

f(x,y) = k exp(-c|x|[sup]a[/sup]-d|y-1|[sup]b[/sup])

might work, though you might need some restrictions on the parameters .

You can probably see these are simple modifications of the Gaussian curve, and you can and “covariance” if you want with xy terms.

Wavelets are a popular choice although they usually do not comply fully with the “n-1 zeros” requirement. (Instead the functions have “ringing tails” with extra zeros.)

This what I was going to say. Actually, any set of odd/even polynomials with 0, 1, 2, 3, …, n zeroes (multiplied by a bell curve) would work; Chebyshev or Legendre polynomials would all work.

The Hermite case is nice because then your test functions obey an orthogonality relationship as well: for any two functions f[sub]n[/sub] and f[sub]m[/sub], the integral of ( f[sub]n/sub*f[sub]m/sub) over x from -∞ to ∞ will vanish unless n = m. But if your application isn’t at the level where you don’t know why this is a good thing, then you probably don’t need to worry about it.

Wavelets are a general term. Perhaps the eigenfunctions others refer to can themselves be considered wavelets?

Yes; I forgot about the part I underlined. For non-orthogonal wavelets, the conditions that f1 and f2 have (n-1) zeros can be easily satisfied I think, i.e. the extra zeros due to “ringing in the tails” can be eliminated… (I can’t remember, or never knew, how convenient it is to satisfy this condition for f3, f4. etc.)

Perhaps surprisingly(*), the best synthesizing filters for signal compression of OP’s form are non-orthogonal. (Surprising in that it may contradict the intuition associated with theorems by Parzival, Karhunen and Loeve.)

Yes I know that Loeve has an accent and that Wachaparzival’s name is not really the same as the Grail hero.

I think it would be more accurate to say that wavelets can be eigenfunctions.