 Solution

ID: past-exam-of-the-analysis-course-of-the-university-of-oxford/2007/b/6/c/solution

Complete the square to obtain, initially for real

y, z

\int_{R} e^{- t^{2}} G (t, y) G (t, z) d t = e^{- y^{2} - z^{2}} \int_{R} e^{- t^{2} + 2 t (y + z)} d t = π e^{2 yz} .

y^{m} z^{n}

on the two sides. The left coefficient is

(m! n!)^{- 1} \int e^{- t^{2}} H_{m} (t) H_{n} (t) d t

, and the right is zero unless

m = n

, in which case it is

π 2^{n} / n!

. The polynomials are real on the real line, so these are the complex

L^{2}

ϕ_{n}

. Thus

⟨ ϕ_{m}, ϕ_{n} ⟩ = δ_{mn} π 2^{n} n!, ∥ ϕ_{n} ∥_{2} = π^{1/4} 2^{n} n! .

Polynomial times Gaussian decay also directly shows each

ϕ_{n}

lies in

L^{2}

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