Editing Hermite polynomials (section)

=== Completeness ===
The Hermite polynomials (probabilist's or physicist's) form an [[orthonormal basis|orthogonal basis]] of the [[Hilbert space]] of functions satisfying
<math display="block">\int_{-\infty}^\infty \bigl|f(x)\bigr|^2\, w(x) \,dx < \infty,</math>
in which the inner product is given by the integral
<math display="block">\langle f,g\rangle = \int_{-\infty}^\infty f(x) \overline{g(x)}\, w(x) \,dx</math>
including the [[gaussian function|Gaussian]] weight function {{math|''w''(''x'')}} defined in the preceding section.

An orthogonal basis for {{math|[[Lp space|''L''<sup>2</sup>('''R''', ''w''(''x'') ''dx'')]]}} is a [[Hilbert space#Orthonormal bases|''complete'' orthogonal system]]. For an orthogonal system, ''completeness'' is equivalent to the fact that the 0 function is the only function {{math|''f'' ∈ ''L''<sup>2</sup>('''R''', ''w''(''x'') ''dx'')}} orthogonal to ''all'' functions in the system.

Since the [[linear span]] of Hermite polynomials is the space of all polynomials, one has to show (in physicist case) that if {{mvar|f}} satisfies
<math display="block">\int_{-\infty}^\infty f(x) x^n e^{- x^2} \,dx = 0</math>
for every {{math|''n'' ≥ 0}}, then {{math|1=''f'' = 0}}.

One possible way to do this is to appreciate that the [[entire function]]
<math display="block">F(z) = \int_{-\infty}^\infty f(x) e^{z x - x^2} \,dx = \sum_{n=0}^\infty \frac{z^n}{n!} \int f(x) x^n e^{- x^2} \,dx = 0</math>
vanishes identically. The fact then that {{math|1=''F''(''it'') = 0}} for every real {{mvar|t}} means that the [[Fourier transform]] of {{math|''f''(''x'')''e''<sup>−''x''<sup>2</sup></sup>}} is 0, hence {{mvar|f}} is 0 [[almost everywhere]]. Variants of the above completeness proof apply to other weights with [[exponential decay]].

In the Hermite case, it is also possible to prove an explicit identity that implies completeness (see section on the [[#Completeness_relation|Completeness relation]] below).

An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for {{math|''L''<sup>2</sup>('''R''', ''w''(''x'') ''dx'')}} consists in introducing Hermite ''functions'' (see below), and in saying that the Hermite functions are an orthonormal basis for {{math|''L''<sup>2</sup>('''R''')}}.