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I'm dealing with a probability problem and I have to understand the following operation on polynomials: let $F$ and $G$ be any two polynomials of variable $p\in [0,1]$ (to be thought of as a Bernoulli parameter).

What can you say about the "interpolated" polynomial $p\cdot F(p)+(1-p)\cdot G(p)$? Is there a standard word denoting this operation?

How can I find references about that? The problem is that when googling "interpolation of polynomials" I mainly find results on polynomial interpolation!

Bonus questions:

  • In my case, the two polynomials induce orientation-preserving homeomorphisms of the closed interval $[0,1]$. Is the resulting interpolation verifying this hypothesis?

  • I can also suppose that $F$ and $G$ both have a unique fixed point other than $0$ and $1$. What about the fixed points of their interpolation?

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For given reals $a,b$, the combination $pa+(1-p)b$ is often called an affine combination, or maybe a convex combination. Maybe those terms would lead to some info. –  coffeemath Feb 15 '13 at 13:19
    
@coffeemath If you make a convex combination, usually you take your barycentric coefficients to be real constants. When I google for "affine combination", results are not what I would like to find! –  Michele Triestino Feb 15 '13 at 14:09
    
The answer to first bonus question is no. Using Weierstrass theorem, one can uniformly approximate any smooth strictly increasing function $f:[0,1]\to [0,1]$ with $f(0)=0$ and $f(1)=1$ by a strictly increasing polynomial that also fixes $0$ and $1$. Using this, we can construct $F$ and $G$ so that $F\le 0.1$ and $G\ge 0.9$ on the interval $[0.1,0.9]$. Then $0.9 F(0.9)+(1-0.9)G(0.9)\le 0.09+0.1=0.19$ while $0.1 F(0.1)+(1-0.1)G(0.1) \ge 0.9^2=0.81$. –  user53153 Feb 15 '13 at 15:02

3 Answers 3

up vote 4 down vote accepted

This question is semi-answerable: nobody can tell for sure that such a thing has not been studied. I am pretty sure that there is no standard name for this. But it's possible that someone, somewhere, sometime wrote down something about this thing. Even so, the chances of you finding it are slim.

In any case, there is no reason to expect this operation to preserve any topological properties. Here are counterexamples to the two questions you asked.

a) Let $F(p)=p^4$, $G(p)=1-(1-p)^4$, and $H(p)=pF(p)+(1-p)G(p)$. The graphs of $F,G,H$ are below: red, blue, green.

Counterexample 1

b) Let $F(p)=\dfrac{p}{19}(27-72p+64p^2)$ and $G(p)=\dfrac{p}{57}(1+120p-64p^2)$. As above, $H(p)=pF(p)+(1-p)G(p)$. The polynomials $F,G,H$ are strictly increasing:

Counterexample 2a

Also, $F$ and $G$ have a unique fixed point other than $0,1$. But $H$ does not. These are the graphs of $F-p,G-p, H-p$: same order of colors as before.

Counterexample 2b

Trading places, we get $pG(p)+(1-p)F(p)$ (not pictured), which is a homeomorphism with two fixed points other than $0,1$.

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A Bernoulli random variable is the same thing as a binomial random with parameter $n = 1$:

Using the Wiener-Askey Polynomial Chaos representation, we know that binomial random variables belong to the Krawchouk-chaos family. That means, if you had, say, a different discrete random variable, then you could represent it as a spectral expansion of a series of Krawchouk polynomials $\Phi_i$, of degree $i$, in a binomial random variable $\zeta$:

$$ y = \sum_{i=0}^\infty y_i \Phi_i(\zeta).$$

By orthogonality of the Krawchouk polynomials, we have

$$\sum_{k=0}^n p^k(1-p)^{n-k}\Phi_r(\zeta)\Phi_q(\zeta) = \frac{(-1)^qq!}{(-n)_q}\left(\frac{1-p}{p}\right)^q \delta_{rq}.$$

But note that $n=1$ for a Bernoulli random variable, so this amounts to

$$p\Phi_r(\zeta)\Phi_q(\zeta) + (1-p)\Phi_r(\zeta)\Phi_q(\zeta).$$

Now, maybe there is something you can perhaps do to factor $F$ and $G$ by Krawchouk polynomials. That might lead to something... I'll have to think a little while on it.

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Thanks to those who gave an answer to my question.

Actually my motivation was in understanding a very specific case.

I eventually managed to find the answer for my initial problem in an old paper by Moore and Shannon Reliable circuits using less reliable relays (look at Theorem 1).

The keyword is reliability polynomials, and many people worked and are working on that.

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