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"If $f(x)$ is a polynomial of degree $m$, show that $f(x)$ may be written in the form $$f(x)=\Sigma_{r=0}^{m}c_rH_r(x),$$ where $$c_r=\frac{1}{2^rr!\sqrt {\pi} }\int_{-\infty}^{\infty}e^{-x^2}f(x)H_r(x)dx.$$ Deduce that $\int_{-\infty}^{\infty}e^{-x^2}f(x)H_r(x)dx=0$ if $f(x)$ is a polynomial of degree less than $n$."

I'm not sure how to get started with this.

J. Hunt
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    Do you know that the Hermite polynomials are a system of orthogonal polynomials with weight $e^{-x^2}$ on $(-\infty,\infty)$? – Chappers Apr 29 '17 at 17:42
  • Yes, in fact applying the orthogonality condition was my first idea. But I don't think $f(x)$ is necessarily a Hermite polynomial. – J. Hunt Apr 29 '17 at 17:45

1 Answers1

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Let $P_m$ be the vector space of polynomials with degree at most $m$. Then $\dim{P_m}=m+1$, since $\{1,x,\dotsc,x^m\}$ is a basis. Since each $H_k$ is a polynomial of degree $m$, $\{H_0,H_1,\dotsc,H_m\}$ must also be a basis (they are linearly independent and spanning by induction: there's no way to make $x^k$ without having a polynomial of degree at least $k$. The first result follows from this.

The second result comes from orthogonality: since $H_m$ is orthogonal to $\{H_0,H_1,\dotsc,H_{m-1}\}$, it is orthogonal to the span of these, $P_{m-1}$.

Chappers
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