# Questions tagged [hermite-polynomials]

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### In what sense does the Hermite expansion of a bounded smooth function converge?

Let $f : \mathbb{R} \to \mathbb{C}$ be a smooth and bounded function. If we denote by $\{ H_n(x) \}$ the sequence of normalized Hermite polynomials, then the Hermite expansion of $f$ is defined as \...
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### Closed formula for Hermite polynomials

Hermite polynomials $H_k(x), x \in \mathbb{R}, k \in \mathbb{N}$ are defined by the formula $$H_k(x)=(-1)^k e^{x^2} \frac{d^k}{d x^k}\left(e^{-x^2}\right) .$$ Each $H_k(x)$ is a polynomial of exact ...
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### Discrete Wavelet Transform and Gaussian decay

I have a question regarding the possibility of constructing a Discrete Wavelet Transform based on a scaling function having Gaussian decay (and no more decay than that). More specifically, I am ...
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### Basis for $L^2(\mathbb{R})$ that Solves the Heat Equation

This is a less-than-serious question that I asked on math.SE, but I suspect it is slightly more appropriate to ask it here. Consider the heat equation $$u_t = \frac12 u_{xx}$$ On $\mathbb{T}$ with ...
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### What are the orthogonal polynomials w.r.t. Maxwell distribution

Is there a way to get a clean presentation of the orthogonal polynomials w.r.t. the Maxwell distribution https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution ? If you separate the ...
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### What are the best known bounds on the Hermite polynomials?

The best I could find on the net is this paper, http://arxiv.org/pdf/math/0401310.pdf Has this been improved?
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### Asymptotic form of $L^1$-norm of Hermite functions
Background Working on a quantum mechanics problem, I've stumbled on the problem of maximizing the functional $$\int_{A} \varphi_m \varphi_n$$ in the limit of large $m$ and $n$, given that $n \gg m$. ...
The inverse of the Weierstrass transform expands a function as a series of Hermite polynomials $H_{n}$. There are several ways to invert the Weierstrass transform which led me to the following ...