Show that Roots of Orthogonal Polynomials are interlaced.

Question

Does anyone have an easy proof to show how the roots of orthogonal polynomials $p_n$ and $p_{n-1}$ are interlaced. I have just shown that all roots of orthogonal polynomials lie in the interval $I$ but that's about it.

$\langle f,g \rangle =\int_If(x)\overline{g(x)}w(x) \, \mathrm{d}x$ where $w(x)$ is the tempered weight function.

Any ideas?

Answer 1

This is a sketch of the proof:

1. Let $P_n$ be the monic orthogonal polynomials for the weight $w(x)$. Show that we have a recurrence

$$P_n(x) = (x-a_n) P_{n-1}(x) - b_{n-1}^2 P_{n-2}(x)$$

Hint: divide $P_n$ by $P_{n-1}$ and show that the remainder is a negative multiple of $P_{n-2}$.

2. Show by induction on $n$ that $P_{n}$, $P_{n-1}$ have interlacing roots. In fact, show the following: if the monic polynomials $P$, $Q$ have interlacing roots ($\deg Q = \deg P-1$), then $(x-a)P- b^2 Q$, and $P$ also have interlacing roots. This you prove as follows: show that $$\frac{Q(x)}{P(x)} = \sum_{i=1}^m \frac{c_i}{x-\alpha_i}$$ where $c_i>0$, so $$\frac{(x-a)P(x) - b^2 Q(x)}{P(x)} = x-a - \sum_{i=1}^m \frac{b^2 c_i}{x-\alpha_i}$$ Sketch the graph of the above rational function and conclude ( see also this answer).

Note: We can also notice that $P_n(x)$ are the characteristic polynomials of the tridiagonal symmetric matrix $A_n$ with bands $(a_1, \ldots, a_n)$ and $(b_1, \ldots, b_{n-1})$, so the conclusion now follows from the interlacing theorem.

Note: From the above we conclude that the roots of the polynomial $P_n(x)$ are real and distinct. It is not hard to show that they lie in the convex hull of the support of the measure $w(x) dx$.