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.

I am trying to prove that the generalized Laguerre polynomials form a basis in the Hilbert space $L^2(\mathbb{R})$. 1. Orthonormality \begin{equation} \int_0^{\infty} e^{-x}x^kL_n^k(x)L_{m}^k(x)dx=\dfrac{(n+k)!}{n!}\delta_{mn} \end{equation} 2. Completeness (?) \begin{equation} \sum_{n=0}^{+\infty}L_n^k(x)L_{n}^k(y)=?\delta(x-y) \end{equation} I am having trouble with the second relation, can anyone give a reference where it is proven or hint for a proof?

share|improve this question
3  
As $L^k_n=x^n/n!+$ lower degree terms, the sequence $L^k_n$, $n=0,1,2,\dots$ can be obtained from $1,x,x^2,x^3,\dots$ by the Gramm-Schmidt orthonormalization process. The completeness is therefore equivalent to the completeness of polynomials in $L^2(\mathbb{R}_+, e^{-x}x^k\,dx)$. –  user8268 Jan 19 '12 at 22:08
    
Of course, $L^2(\mathbb{R})$ should read $L^2(\mathbb{R}_+)$. –  ˈjuː.zɚ79365 Jun 12 '13 at 1:10

1 Answer 1

Completeness of an orthogonal sequence of functions is a bit tricky on unbounded intervals, while it is relatively straightforward on bounded intervals. In the case of Laguerre and Hermite polynomials, there is a nice trick due to von Neumann that allows the reduction to bounded intervals.

There seems to be a bit of confusion about the interval in the statement of the question. Here's a correct statement:

For any real number $\alpha \gt -1$ the functions $\langle e^{-x/2} x^{\alpha/2} L_{n}^{(\alpha)}(x)\rangle_{n=0}^\infty$ obtained from the Laguerre polynomials $L_{n}^{(\alpha)}(x)$ are a complete orthogonal system in $L^2(0,\infty)$. The Hermite polynomials $H_n(x)$ yield the complete orthogonal system $\langle e^{-x^2/2} H_n(x)\rangle_{n=0}^\infty$ in $L^2(\mathbb{R})$.

This is proved in detail in the classic book Gábor Szegő, Orthogonal polynomials, Chapter 5. The entire chapter discusses the main properties oft the Laguerre polynomials $L^{(\alpha)}_n(x)$ for an arbitrary real number $\alpha \gt -1$ and proves their completeness in Section 5.7.

More precisely, Szegő shows in Theorem 5.7.1 on pages 108f that for fixed $\alpha \gt -1$ the functions $f_n(x) = e^{-x/2}x^{\alpha/2} x^n$ span a dense subspace of $L^2(0,\infty)$.

The first idea is to use a change of variables $y = e^{-x}$ in order to use the case of $L^2(0,1)$ where density of the span of $(\log1/y)^{\alpha/2} y^n$ is not too hard to prove (see Theorem 3.1.5).

Write a function in $L^2(0,\infty)$ as $e^{-x/2} x^{\alpha/2} f(x)$. Then we have that $(\log1/y)^{\alpha/2} f(\log(1/y)) \in L^2(0,1)$ can be approximated by functions of the form $(\log1/y)^{\alpha/2} p(y)$ where $p$ is a polynomial. Transforming back to $(0,\infty)$ this shows that $$ \int_{0}^\infty e^{-x} x^\alpha (f(x) - p(e^{-x}))^2 \,dx \lt \varepsilon $$ for a suitable polynomial $p$. This reduces the task to proving that for all natural $k$ there exists a polynomial $q$ such that $$\tag{$\ast$} \int_{0}^\infty e^{-x} x^\alpha (e^{-kx} - q(x))^2\,dx $$ is as small as we wish.

To do this, von Neumann's trick is to use the generating function of the Laguerre polynomials $L_{n}^{(\alpha)}(x)$ $$ (1-w)^{-\alpha-1} \exp\left(-\frac{xw}{1-w}\right) = \sum_{n=0}^\infty L_n^{(\alpha)}(x) w^n. $$ Choosing $w = \frac{k}{k+1}$ we have $\exp\left(-\frac{xw}{1-w}\right) = \exp{(-kx)}$.

Thus, a natural choice for $q$ is $q_N(x) = (1-w)^{\alpha+1} \sum_{n=0}^N L_n^{(\alpha)}(x) w^n$ with large enough $N$. Plugging this into $(\ast)$ we obtain using the orthogonality relations $$ \begin{align*} \int_{0}^\infty e^{-x} x^\alpha (e^{-kx} - q_N(x))^2\,dx & = (1-w)^{2\alpha+2} \int_{0}^\infty e^{-x} x^\alpha \left(\sum_{n=N+1}^\infty L_{n}^{(\alpha)}(x) w^{n}\right)^2\,dx \\ &= (1-w)^{2\alpha+2} \Gamma(\alpha+1) \sum_{n=N+1}^\infty \binom{n+\alpha}{n} w^{2n} \end{align*}$$ where term-wise integration is justified using an application of Cauchy-Schwarz. It remains to observe that the last expression tends to $0$ as $N \to \infty$.

Another reference discussing the case of $\alpha = 0$ nicely is Courant and Hilbert, Methods of mathematical physics, I, §9, sections 5 and 6. They discuss ordinary Laguerre and Hermite polynomials and their completeness.

share|improve this answer

Your Answer

 
discard

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.