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 \...
Isaac's user avatar
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41 views

Lower bound for the fractional Sobolev norm of the Hermites function

For $r \in (0, 2)$, I am interested in a lower bound for the quantity : $$I_r(n) := \int_{\mathbb{R}} |f_n(x)|^2 |x|^{r} dx$$ where $f_n(x) = (-1)^n (\sqrt{\pi} n! 2^n)^{-1/2} e^{x^2/2} \dfrac{d^n}{dx^...
jvc's user avatar
  • 183
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0 answers
85 views

Closed formula for Laguerre

Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$. Assume $0<\beta<1$. Is there a closed formula for this sum $$\sum^\infty_{j=0} \frac{1}{(b+j)^{1-\beta}}L^{m}_j(x)$$ where $b>0$ and $...
zoran  Vicovic's user avatar
0 votes
1 answer
153 views

A question regarding Hermite polynomials and exponential operators $\exp[e^{x^2/2}p(\frac{d}{dx})e^{-x^2/2}]f(x)$

Is it possible to express $$\exp\left[\mathrm{e}^{x^2/2}p\left(\frac{d}{dx}\right)\mathrm{e}^{-x^2/2}\right] f(x)$$ as an integral transform or something similar? $p(x)$ is a polynomial. $$\exp\left[\...
Mirar's user avatar
  • 350
3 votes
0 answers
60 views

Closed formula for $\sum\limits^\infty_{k=0}\frac1{(k+a)(k+b)} L^1_k(x)L^1_k(y) $

Let $ L^{\alpha}_{n}(x)=\sum^{n}_{k=0} \binom{n+\alpha}{n-k}\big(-1\big)^{k}\frac{x^{k}}{k!},\alpha>-1$ be Laguerre polynomials of type $ n$. Is there a closed formula for $$\sum^{\infty}_{k=0}\...
zoran  Vicovic's user avatar
1 vote
2 answers
269 views

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 ...
zoran  Vicovic's user avatar
2 votes
0 answers
121 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-...
JayanthJ's user avatar
5 votes
2 answers
394 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 ...
xiangsha's user avatar
2 votes
2 answers
258 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 ...
dohmatob's user avatar
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1 vote
1 answer
1k 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 ...
Gabe's user avatar
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2 votes
0 answers
114 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 ...
Andrius Kulikauskas's user avatar
3 votes
2 answers
270 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 $w\in\mathbb{C}$...
Yonah Borns-Weil's user avatar
2 votes
2 answers
911 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 ...
Bazin's user avatar
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1 vote
1 answer
391 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. ...
miathsboi's user avatar
3 votes
0 answers
212 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 ...
Abhishek Panigrahi's user avatar
2 votes
1 answer
365 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 ...
Abhishek Panigrahi's user avatar
2 votes
1 answer
800 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)...
Jackie Lu's user avatar
  • 339
1 vote
0 answers
105 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 ...
S. Montaner's user avatar
8 votes
0 answers
460 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 ...
Greg Zitelli's user avatar
1 vote
2 answers
504 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^...
Z. Alfata's user avatar
  • 640
4 votes
0 answers
160 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, ...
Paata Ivanishvili's user avatar
12 votes
1 answer
494 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 ...
David Lehavi's user avatar
  • 4,299
3 votes
1 answer
2k 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?
gradstudent's user avatar
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4 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 $n \gg m$. ...
Mateus Araújo's user avatar
5 votes
0 answers
685 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 ...
Craig Calcaterra's user avatar