Questions tagged [hermite-polynomials]

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2 votes
0 answers
82 views

Evaluation of a summation involving Hermite polynomials

I started with the the following four-variable function, $f(s,x,y,u)$, expressed as a summation involving the product of Hermite polynomials. $f(s,x,y,u)=\sum_{n=0}^{\infty}\frac{H_{n}(x)H_{n}(y)}{(n-...
5 votes
2 answers
364 views

How to integrate the multinomial over a ball in $\mathbb{R}^{n}$?

I got an interesting question. Consider this integral: $$ \int_{B(0,1)}\bigg(\sum_{j=1}^{n}a_{j}x_{j}^2\bigg)^m \mbox{d}x, \quad m,n\in \mathbb{N}, \ a_{i}>0, \ i=1,2,\ldots,n.$$ It is clear that ...
2 votes
2 answers
182 views

Integral of product of Hermite polynomials w.r.t marginal distribution of first two-coordinate of random vector on unit-sphere

This question is related to: https://math.stackexchange.com/q/4270522/168758 Let $H_n(x) \in \mathbb R[x]$ be the probabilist's $n$th Hermite polynomial. This an $n$th degree polynomial given by the ...
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1 vote
1 answer
655 views

Integrating a B-Spline basis function with respect to the standard normal PDF

I am looking for ways to evaluate exactly (i.e. analytically or semi-analytically) integrals of the type: $$ \int_{-\infty}^{+\infty}B_{i}^k(u)e^{-\frac{(u-\mu)^2}{2\sigma^2}}du, $$ where $B_i^k$ is a ...
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2 votes
0 answers
103 views

Deduce Sheffer's classification of orthogonal polynomials of A-type 0

Theorem 1.9 in Daniel Galiffa and Tanya Riston's paper, An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences, 2015, presents without proof Isador Sheffer's ...
3 votes
2 answers
181 views

Complex Hermite polynomial orthogonality on weighted space

Consider the "probabilist's" Hermite polynomials given by $$H_n(x)=(-1)^ne^{\frac{x^2}{2}}\partial_x^ne^{-\frac{x^2}{2}}.$$ These polynomials trivially extend to functions of $w\in\mathbb{C}$...
2 votes
2 answers
625 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 ...
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0 votes
1 answer
348 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
164 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
281 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
707 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)...
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1 vote
0 answers
90 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
423 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
468 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^...
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4 votes
0 answers
147 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, ...
12 votes
1 answer
431 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 ...
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3 votes
1 answer
1k 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?
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3 votes
1 answer
1k 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
672 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 ...