Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question
Try the Gram-Schmidt process on the basis $\{1, x, x^2,\ldots \}$. –  Christopher A. Wong May 29 '13 at 5:31
add comment

1 Answer

up vote 1 down vote accepted
  • $n = 0$: $\phi_n$ 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.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.