Questions tagged [hermite-polynomials]

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Polynomial mean [closed]

I am not able to justify the above inequality, contained in Garcia-Cuerva, Rubio De Francia "Weighted Norm Inequalities and Related Topics", pag.292-293, at the end of 292. Let $P_N(f)$ be ...
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1answer
31 views

Polynomial Interpolation $p(0)=1$ and $p'(-1)=p(1)$

The following practice problem was given: Find all polynomials $p(x)$ that satisfy the interpolation conditions: $p(0)=1$ $p'(-1)=p(1)$ and the degree of $p(x)$ as small as possible. I don't know ...
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1answer
60 views

Do Hermite polynomials satisfy $\int(H_n(x))^2 e^{-x^2} dx= 2n \int (H_{n-1}(x))^2 e^{-x^2} dx$?

I'm trying to prove that the norm of the Hermite polynomials (physicist's version) equals $ 2^n n! $. I stumbled upon this answer and I don't understand parts of the proof. First of all, it seems ...
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14 views

Gaussian Hermite expansion for complex numbers

There exists a way to expand a gaussian function into a series of Hermite polynomials as $$ \sum_{n=0}^{\infty}r^{n}\left[H_{n}(x)\right]^{2} = \frac{1}{\sqrt{\pi(1-r^{2})}}\exp\left(\frac{2r}{1+r}x^{...
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37 views

Suitable choices of base functions for periodic signals using Koopman approximation

This article explains the extended dynamic mode decomposition (EDMD) algorithm. It is about the spectral decomposition of nonlinear dynamics based on the Koopman operator theory. Particularly, it is ...
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10 views

Third-order grid-characteristic approximation scheme for the transport equation on a three-point template

A mixed problem for a linear one-dimensional transport equation is considered: \begin{equation*} \begin{cases} \dfrac{\partial u}{\partial t} + a\dfrac{\partial u}{\partial x} = 0,& 0 < x\leq ...
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1answer
22 views

Nonparametric Hermite cubic to Bezier Curve

I have a Hermite cubic in the form of the standard cubic equation of $y=ax^3+bx^2+cx+d$, and this works well for my interpolation needs, but now I want to create Bezier curves from the Hermite cubics. ...
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2answers
99 views

Hermite polynomials (Integral)

Could you please help me? How to evaluate this integral $$\int_{-\infty }^{\infty }x e^{-x^2}H_{2n-1}(xy)dx$$ I tried to use a recurring formula like: $$H_{2n-1}(xy)=2xyH_{2n-2}(xy)-2(2n-2)H_{2n-3}(xy)...
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23 views

Asymptotic expansion of scaled Hermite polynomial

I have a scaled probabilist's Hermite polynomial: \begin{equation} H_n(ax)e^{-bx^2} \end{equation} where $a$ and $b$ are both complex numbers. What is the asymptotic limit of this expression as $n \to ...
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2answers
22 views

closed form for arbitrary derivatives of gaussian function

What is the closed form expression for the $n$th order derivative of the Gaussian function? $$ \frac{d^n}{dx^n}\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}}$$ Wikipedia says ...
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1answer
27 views

Are the Hermite polynomials independent?

I believe they are not independent as obviously, $$X := He_1(x) = x \; \text{and} \; Y:=He_2(x) = x^2-1$$ then $$P(Y\in A | X \in A) \neq P(Y \in A) \; \; \text{for some set } A \subset \mathbb{R}.$$ ...
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13 views

Piece-wise Hermite Cubic Polynomial

If we know points x0,x1,x2, as well as f(x) and f'(x) for each point, is it possible for the cubic Hermite interpolating polynomial to have a continuous second derivative?
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1answer
26 views

Integration with unbounded spectral weight

I want to consider the solution of an equation in the space $L^2(e^{-2\alpha x} \mathbb{R}. )$ I am just wondering how to implement the weight when computing the norm. So far, I use Hermite spectral ...
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43 views

Help with the Hermite differential equation

I am studying the differential Hermite equation for two different books. And I noticed that they have a little difference in the expression for Hermite's EDO. I'll show you: The first author I am ...
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33 views

How to show Hermite polynomials associated with standard Gaussian measure are not independent?

I would like to use Weak Law of Large Numbers to prove by contradiction that if they are mutually independent then $\frac{1}{n^2}\sum_{i=1}^{n}\textbf{E}(X_i^2)\rightarrow0$ as $n\rightarrow \infty$, ...
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34 views

Hermite polynomials from binomial theorem

Hello I want to obtain the explicit formula for the Hermite polynomial: $$ H_n(x)=\sum_{k=0}^{[n/2]}\frac{(-1)^k n!}{k! (n-2k)!} (2x)^{n-2k} $$ by expanding the exponential in the generating function $...
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43 views

How to calculate Hermite interpolation without slope in the end points (unknown derivatives)?

In class, the following exercise was proposed. Determine the natural cubic spline that passes through the points (0,1) (1, 0) (2, 0.5) (3, 1). Use the Hermite method I need to interpolate a function, ...
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14 views

How to find weight and points of half interval Gauss Hermite quadrature?

I want to find weight $w_i$ and points $x_i$ such that $$\int_0^\infty \exp (-x^2)f(x)dx \approx \sum_{i=1}^n w_if(x_i)$$ Is there useful reference for this? I'm implementing half interval Gauss-...
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29 views

Term-by-term differentiation for the generating function of the Hermite polynomial

Hermite polynomial $H_n$ is defined by $$H_n(x) := \frac{(-1)^n}{n!}e^{x^2/2} \frac{d^n}{dx^n}e^{-x^2/2} $$ and its generating function is given by $F(t,x) := \exp{(tx-t^2/2)} $, i.e. $$\sum_{n=0}^\...
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52 views

Construction of Hermite Polynomials

I am studying Hermite Interpolation and the most common practice I came across was the use of Lagrange Polynomials. I tried to construct a Hermite Polynomial for 2 points $x_0$ and $x_1$ but instead I ...
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0answers
34 views

Prove that $\int_{-\infty}^{\infty} e^{-x^2} H_m(x)H_n(x)dx$ [duplicate]

$\int_{-\infty}^{\infty} e^{-x^2} H_m(x)H_n(x)dx=\left\{\begin{matrix} 0, & m \neq n\\ 2^n n! \sqrt{\pi},& m = n \end{matrix}\right.$ Thats a Hermite polynomials. I need a hint, i really ...
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15 views

Gaussian expectation of Lagrange polynomial

Let $\{x_i\}_{i=1}^N$ be the zeros of the probabilists' Hermite polynomial and $L_j(x)$ for $j = 1, \ldots, N$ the Lagrange polynomials with nodes $\{x_i\}_{i=1}^N$. Moreover, let $\rho \in [0, 1]$ ...
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1answer
32 views

Expanding Dawson's integral in a series of Hermite functions

We want to show that $$F(x)=\sqrt{\pi } \sum _{n=0}^{\infty } \frac{(-1)^n H_{2 n+1}(x)}{2^{3 n+3} \Gamma \left(n+\frac{3}{2}\right)}$$ where $F(x)$ is the Dawson Integral ($F(x)=\exp \...
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1answer
43 views

Integral Involving Hermite Polynomial and rational function

We want to prove the formula $$\frac{\int_{-\infty }^{\infty } \frac{e^{-x^2} H_n(x){}^2}{1+x^2} \, dx}{2^n n!}=\int_{-\infty }^{\infty } \frac{\left(\frac{1-x^2}{1+ x^2}\right)^n ...
3
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1answer
99 views

Formulas's deduction from the generating function of Hermite polynomials

In the book "Essential Mathematical Methods for Physicists" comes the following problem that I am trying to solve: at first I could see that the first formula that is given as an answer is ...
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34 views

How is the formula $H_{2n+1}(x)=(-1)^n\sum_{s=0}^{n}(-1)^s(2x)^{2s+1}\cfrac{(2n+1)!}{(2s+1)!(n-s)!}$ derived?

How the relationship $H_{2n+1}(x)=(-1)^n\sum_{s=0}^{n}(-1)^s(2x)^{2s+1}\cfrac{(2n+1)!}{(2s+1)!(n-s)!}$ is deduced ,working with Hermite polynomials ? There is a problem so I don't know how to deduce ...
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27 views

Proving that the $2n$ Hermite polynomial is $H_{2n}(x)=(-1)^n\sum_{s=0}^{n}(-1)^{2s}(2x)^{2s}\cfrac{(2n)!}{(2s)!(n-s)!}$

I want to prove that the $2n$ Hermite polynomial is $H_{2n}(x)=(-1)^n\sum_{s=0}^{n}(-1)^{2s}(2x)^{2s}\cfrac{(2n)!}{(2s)!(n-s)!}$ So I resort to equality: $\begin{equation*} H_n(x)=\sum_{s=0}^{n/2}(-1)^...
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0answers
54 views

Problem proving $\int_{-\infty}^{\infty}x^2e^{-x^2}H_n(x)H_m(x)dx$

I am trying to show that the value of the following integral is: $$\int_{-\infty}^{\infty}x^2e^{-x^2}H_n(x)H_m(x)dx=2^{n-1}\pi^{\frac{1}{2}}(2n+1)n!\delta_{mn}+2^n\pi^{\frac{1}{2}}(n+2)!\delta_{(n+2)m}...
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0answers
44 views

Proving that $\int_{-\infty}^{\infty}x^{m}e^{-x^2}H_n(x)dx=0$ for $m$ an integer and $0\leq m\leq n-1$

I find myself trying to prove that $\int_{-\infty}^{\infty}x^{m}e^{-x^2}H_n(x)dx=0$ for $m$ an integer and $0\leq m\leq n-1$ I tried to solve it by the following method: Using the Rodrigues formula ...
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1answer
53 views

Spectrum and point spectrum of quantum harmonic oscillator

I have this question concerning the Hermite operator on $\mathbb{R}^n$ (defined by $Hf:=-\Delta f+|x|^2f$). We know that $H$ is a densely-defined unbounded operator on $L^2(\mathbb{R}^n)$ and that $\...
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1answer
81 views

Prove $f(\xi)$ behaves as $e^{\xi^2}$ if $\lambda<0$ in $\frac{d^2f}{d\xi^2}-2\xi\frac{df}{d\xi}+2\lambda f(\xi)=0$

How does one show that $f(\xi)$ will behave as $e^{\xi^2}$ if $\lambda<0$ in the Hermite differential equation $\frac{d^2f}{d\xi^2}-2\xi\frac{df}{d\xi}+2\lambda f(\xi)=0$ This problem comes in ...
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2answers
58 views

General solution for an integral with Hermite polynomials?

I'm currently studying quantum mechanics and thus dealing with Hermite polynomials. By looking on Wikipedia I found this identity for Hermite polynomials: $$\int\limits_{-\infty}^{\infty} H_m(x)H_n(x)...
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1answer
26 views

Help me to show that exist only one interpolation polynomial with such properties

Let $f:[a,b]\to \Bbb R$, $f \in C^2[a,b]$ and $x_0,x_1,x_2 \in [a,b]$ where $ x_0 \neq x_2$. Show that exists only one polynomial $p \in P_3$ for which $$p(x_0)=f(x_0),\ p\prime(x_1)=f\prime(x_1),\ ...
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0answers
26 views

Using induction to prove a question based on Hermite polynomials [duplicate]

I am practising for a test that I have next week on proofs. I have tried to do this question using induction, but it is quite different to other induction questions I have done. Any guidance/...
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1answer
67 views

Integral representation of Hermite polynomial

I'm studying Merzbacher's Quantum Mechanics. In his treatment of harmonic oscillators, he introduces Hermite polynomials and their properties. He mentions that the Hermite polynomials (in the form ...
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0answers
49 views

I am unsure about this question using hermite polynomials and satifying recurrence relations

How would I answer a question like this? It is in a practise question for a test.
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1answer
81 views

Integral relation between Hermite and Laguerre polynomials

I'd like to proove the following integral relation $$ \frac{1}{2^m m!} \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty}\,\mathrm{d}\zeta \, e^{-\zeta^2} H_m(\zeta+\zeta_1)H_m(\zeta+\zeta_2) = L_m(-2\...
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3answers
172 views

Second order inhomogeneous equation: $y''-2xy'-11y=e^{-ax}$

My question relates to an a second order inhomogeneous equation: $$y''-2xy'-11y=e^{-ax}$$ First I need to investigate the homogeneous equation: $$y''-2xy'-11y=0$$ $$y''-2xy'=11y$$ Forms Hermite's ...
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0answers
44 views

bicubic Hermite spline interpolation: How to calculate the cross derivative?

I want to implement the bicubic interpolation of a hermite spline (specifically the akima spline). That means I have the points, and the derivatives in x & y direction for each point. But I do not ...
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0answers
56 views

Hermite polynomial relations

How one can prove the relation on Hermite polynomial given as $$\int_{-\infty}^\infty H_n\left(x+\frac{x_0}{2}\right)e^{^{-\frac{x^2}{2}}}dx=\sqrt{\pi}x_0^n$$I also didn't understand the meaning of $\...
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2answers
116 views

Proving $\frac{1}{e}\cosh(2x)=\sum_{n=0,1,2,…}^\infty\frac{1}{(2n)!}H_{2n}(x)$, where $H_{2n}(x)$ is the even order Hermite polynomial

How can we prove the following relation $$\frac{1}{e}\cosh(2x)=\sum_{n=0, 1, 2, ...}^\infty\frac{1}{(2n)!}H_{2n}(x)$$ where the $H_{2n}(x)$ is the even order Hermite polynomial? I didn't get any clue ...
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1answer
62 views

Hermite polynomial generating function

How would I write the following polynomial in terms of the Hermite polynomials, $H_n(z)$? \begin{equation} P_n(z) = \sum_{k=0}^{[n/2]} \frac{n!a^k}{k!(n-2k)!}(2a z)^{(n-2k)} \end{equation} I have ...
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0answers
53 views

integral of hermite polynomials

Objective: Show that $$ \int^{\infty}_{-\infty} x e^{-x^2} H_n(x) H_m(x) dx = \pi^{1/2} 2^{n-1} n! \delta_{m,n-1} + \pi^{1/2} 2^n (n+1)! \delta_{m,n+1} $$ My attempt at this is: \begin{eqnarray*} \...
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1answer
87 views

Fourier transform of Hermite polynomial via generating function

I've spent a lot of times trying to show that $$ \mathcal{F}[e^{-x^2/2} G(x,t)] = e^{-k^2/2} G(k, -it) $$ with $G(x,t)$ being the generating function of Hermite polynomial, $$ G(x,t) = e^{2tx - t^2} $$...
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1answer
72 views

What is the difference between probabilistic and physicists Hermite polynomials?

In the equation they differ only by a fraction of 2 and in the results one is without a constant in x^n and one has those constants. Why are they defined differently and what are their separate uses?
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1answer
81 views

Hermite - Interpolation

This is my first question here. I hope I can find an answer to my question. I tried to find the answer in Books, Videos, Scripts and german forums (I'm german). But nobody could help me. It's about ...
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3answers
393 views

Solving $\left(x-c_1\frac{d}{dx}\right)^nf(x)=0$ for $f(x)$

I'm given that $$\left(x-c_1\frac{d}{dx}\right)^nf(x) = 0$$ I have to solve for $f(x)$ in terms of $n$. For $n=0$: $$f(x)=0 \tag{0}$$ For $n= 1$: $$\begin{align} xf(x) - c_1f'(x) &= 0 \\ \quad\...
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1answer
54 views

New Hermite polynomial identity? (Vol II)

I am trying to prove an identity involving Hermite polynomials using other identities from Wikipedia, but I can't find the way. I have checked the identity in Mathematica for many values of $n$ and it ...
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0answers
36 views

How to find the general solution using the Hermite polynomial?

I want to solve this equation $${\frac{1}{(\sqrt{2m})^n}\Bigr(\frac{\hbar}{i}\frac{d}{dx} - im\omega x\Bigr) }^n\psi(x) =0$$ See more about Hermite function recursion relation $(x-\partial) (e^{x^2/2}\...
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1answer
109 views

New Hermite polynomial identity?

I am trying to prove an identity involving Hermite polynomials using other identities from Wikipedia, but I can't find the way. I have checked the identity in Mathematica for many values of $n$ and it ...