# Difference Equations and displacement operator

For a Prep exam

Exercise from the book: Numerical analysis of scientific computing. Section 1.3-3

Let $p$ be a polynomial of degree $m$, with $p(0) \neq 0$. If a sequence $x$ contains $m$ consecutive zeros and $p(E)x = 0$, then $x=0$

Where:

$E$ denotes the displacement operator, $Ex=[x_2, x_3, \dots]$ where $x = [x_1, x_2, \dots]$

$L: V \to V$

$L=P(E) = \sum_{i=0}^nc_i E^i$ where $P(\lambda) = \sum_{i=0}^m c_i \lambda^i$

I've tried:

I have studied the definitions of from Kincaid's book and I have found two theorems that says that:

$\textbf{Theorem 1:}$

Let $p$ be a polynomial satisfying $p(0) \neq 0$. Then a basis for the null space of $p(E)$ is obtained as follows. With each zero $\lambda$ of $p$ having multiplicity $k$, associate the $k$ basic solutions $x(\lambda), x^{'}(\lambda), \dots, x^{k-1}(\lambda)$, where $x(\lambda) = [\lambda, \lambda^2, \lambda^3, \dots]$.

$\textbf{Theorem 2:}$

For a polynomial p satisfying $p(0) \neq 0$, these properties are equivalent:

(i) The difference equation $p(E)x = 0$ is stable

(ii) All zeros of $p$ satisfy $|z| \leq 1$, and all multiple zeros satisfy $|z|< 1$

but I don't know how to start, at the beginingI was thinking to show that $P(E)=\sum_{i=0}^n c_iE^i /neq 0$, but I am not sure if this is the way, any suggestions?

Thank u

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What is $E$? ${}{}{}{}$ –  copper.hat Nov 6 '13 at 17:20
$E$ denotes de displacement operator $Ex=[x_2, x_3, \dots]$ where $x = [x_1, x_2, \dots]$ –  Louis Nov 6 '13 at 17:24