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Hermite polynomials and Schwartz spaceby Anden
Tags: hermite schwartz 
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#1
Jan2012, 07:00 AM

P: 76

1. The problem statement, all variables and given/known data
I'm supposed to show that the Hermite Polynomials are in Schwartz space [itex]h_n = \frac{1}{\sqrt{n!}}(A^{\dagger})^n h_0[/itex] where [itex]A^{\dagger} = \frac{1}{\sqrt{2}}(\frac{d}{dx} + x)[/itex] and [itex]h_0 = \pi^{1/4}e^{x^2/2}[/itex] 2. Relevant equations Seminorm: [itex]\\phi\_{\alpha,\beta} = \sup_{x\in\mathbb{R}^n}x^{\alpha}\partial^{\beta}\phi(x)[/itex] ([itex]\alpha[/itex] and [itex]\beta[/itex] are multiindices) Schwartz space is defined as being the set [itex]S(\mathbb{R}^n) = \{\phi\in C^{\infty}(\mathbb{R}^n)\\phi\_{\alpha,\beta}< \infty \ \forall\alpha,\beta\in\mathbb{N}^n\}[/itex] 3. The attempt at a solution My idea was to show that [itex]A^{\dagger}: S(\mathbb{R})\rightarrow S(\mathbb{R})[/itex] is a linear map from Schwartz space to itself. Then it would be enough to show that [itex]h_0[/itex] is in Schwartz space for the problem to be solved Multiplication with [itex]x[/itex] and [itex]d/dx[/itex] are linear maps from Schwartz space to itself since according to the definition of the seminorm [itex]\\cdot\_{\alpha,\beta},\ \alpha,\beta\in\mathbb{Z}_{>0}[/itex] [itex]\x\phi(x)\\leq \\phi(x)\_{\alpha+1,\beta} + \beta\\phi(x)\_{\alpha,\beta1}[/itex] and [itex]\\frac{d}{dx}\phi(x)\ = \\phi(x)\_{\alpha,\beta+1}[/itex]. This means that the first part is finished. Is this approach valid or have I completely misunderstood the concept? Also, how do I show that [itex]e^{x^2/2}\in S(\mathbb{R})[/itex]? 


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