# Questions tagged [hermite-polynomials]

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13
questions

**2**

votes

**2**answers

213 views

### On Mehler's formula for Hermite polynomials

In the reference article of Richard Askey and George Gasper published in the American Journal of Mathematics, Autumn, 1976, Vol. 98, No. 3 (Autumn,1976), pp. 709-737, they attribute on page 731 the ...

**0**

votes

**1**answer

204 views

### Poisson Summation Formula appears to fail when applied to Hermite Functions (why?) [duplicate]

I came across an odd circumstance where it appears as though the poisson summation formula fails to yield a correct answer (involving Hermite Functions), and I don't quite understand why this happens. ...

**3**

votes

**0**answers

107 views

### Lower bound on coefficients in hermite transform of Tanh

I would like to understand $L^2\left(\mathbb{R}, \mu\right)$ approximation by polynomials of $\tanh$ (and more generally smooth functions), where $\mu$ is standard gaussian distribution. This leads to ...

**1**

vote

**1**answer

188 views

### Hermite Transform of Tanh

I would like to understand $L^2(\mathbb{R},\mu)$ approximation by polynomials of $\tanh$ up to degree $n$ where $\mu$ is the standard Gaussian distribution. This leads to considering the Hermite ...

**2**

votes

**1**answer

343 views

### A special solution to the Hermite Differential Equation

I know that the general form solution to the Hermite differential equation
$$ y''-2xy'+2\lambda y=0$$
is
$$y(x)=a_1 M(-\frac{\lambda}{2},\frac{1}{2},x^2)+a_2 H(\lambda,x),$$
where $M(\cdot,\cdot,\cdot)...

**1**

vote

**0**answers

52 views

### 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 ...

**8**

votes

**0**answers

306 views

### 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 ...

**1**

vote

**2**answers

280 views

### Generating function for products of complex Hermite polynomials

By making use of the generating function
$$\sum_{m=0}^\infty \frac{H_m(x)}{m!} t^m=e^{-t^2 + 2xt} $$ for the real Hermite polynomials $H_m$, we get easily that
$$(*)\quad \sum_{m,n=0}^\infty \frac{u^...

**4**

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**0**answers

112 views

### Turán's inequalities for Hermite functions

Given $\lambda \in \mathbb{R}$ let $H_{\lambda}(x)$ be the solution of the Hermite differential equation:
$$
\frac{d^{2}}{dx^{2}} H_{\lambda}(x)-x\frac{d}{dx}H_{\lambda}(x)+\lambda H_{\lambda}(x)=0, ...

**11**

votes

**1**answer

354 views

### 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 ...

**3**

votes

**1**answer

887 views

### 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?

**3**

votes

**1**answer

859 views

### 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 $n \gg m$. ...

**5**

votes

**0**answers

647 views

### Has the Weierstass transform been used to give Hermite series representations of the Riemann zeta function?

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 ...