Inner product polynomials

Question

Let $V$ be the vector space of real polynomial $\mathbb{R}[x]$ endowed with the inner product

$\langle f,g \rangle = \displaystyle\int_{-\infty}^{\infty} e^{-|x|}f(x)g(x) \ dx$

By considering the sequence of subspaces $\{V_n\}$ where

$V_n = \{f(x) \in \mathbb{R}[x] : \deg f \leq n \}$

or otherwise, show that there exist unique monic polynomials $\phi_n(x)$ for $n \geq 0$ such that

$\displaystyle\int_{-\infty}^{\infty} e^{-|x|}\phi_n(x)g(x) \ dx = 0$

whenever $\deg g < n$, and find $\phi_n(x)$ for $n = 0,1,2.$

What is the coefficient of $x^{2000}$ in $\phi_{2007}(x)$?

I'm having trouble even knowing where to begin with this question, any help appreciated!

Answer 1

• $n = 0$: $\phi_n$ being monic implies $\phi_0 = 1$, if $\mathrm{deg}(g)<0$, then $g = 0$, and $$ \int_{-\infty}^{\infty} e^{-|x|}\phi_0(x)g(x) \, dx = 0 $$ is trivial.

• $n= 1$: $\phi_1$ has to be $x$ for being monic, and $g = c$ is a constant, by symmetry of the integral: $$ \int_{-\infty}^{\infty} e^{-|x|}\phi_1(x) c\, dx = 0 $$ for $e^{-|x|}\phi_1(x)$ is odd.

• $n= 2$: let $\phi_2(x) = x^2 + a_1 x +a_2$, $g\in \mathrm{span}\{1,x\}$, by symmetry $$ \int_{-\infty}^{\infty} e^{-|x|}(x^2 + a_1 x +a_2) x\, dx = 0 $$ implies $a_1= 0$, and $$ \int_{-\infty}^{\infty} e^{-|x|}(x^2 + a_2)\, dx = 0 $$ yields $a_2 = -2$. Hence $\phi_2(x) = x^2 - 2$ which does not have a degree 1 term.

• $n= 2007$: $g\in \mathrm{span}\{1,x,\ldots,x^{2006}\}$, choosing certain $g$ will let you find out certain degree terms in $\phi_{2007}$ is zero by a symmetry argument like above (the integration of an odd function is zero from $-\infty$ to $\infty$, given that it is integrable), the rest is left for you to try.

• General result for $\phi_n$, discuss the cases for $n$ being odd and even like above.