Questions tagged [hermite-polynomials]
The hermite-polynomials tag has no usage guidance.
171
questions
1
vote
0answers
30 views
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 ...
0
votes
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 ...
3
votes
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 ...
1
vote
0answers
14 views
Gaussian Hermite expansion for complex numbers
There exists a way to expand a gaussian function into a series of Hermite polynomials as
$$ \sum_{n=0}^{\infty}r^{n}\left[H_{n}(x)\right]^{2} = \frac{1}{\sqrt{\pi(1-r^{2})}}\exp\left(\frac{2r}{1+r}x^{...
2
votes
0answers
37 views
Suitable choices of base functions for periodic signals using Koopman approximation
This article explains the extended dynamic mode decomposition (EDMD) algorithm. It is about the spectral decomposition of nonlinear dynamics based on the Koopman operator theory.
Particularly, it is ...
0
votes
0answers
10 views
Third-order grid-characteristic approximation scheme for the transport equation on a three-point template
A mixed problem for a linear one-dimensional transport equation is considered:
\begin{equation*}
\begin{cases}
\dfrac{\partial u}{\partial t} + a\dfrac{\partial u}{\partial x} = 0,& 0 < x\leq ...
0
votes
1answer
22 views
Nonparametric Hermite cubic to Bezier Curve
I have a Hermite cubic in the form of the standard cubic equation of $y=ax^3+bx^2+cx+d$, and this works well for my interpolation needs, but now I want to create Bezier curves from the Hermite cubics. ...
1
vote
2answers
99 views
Hermite polynomials (Integral)
Could you please help me? How to evaluate this integral
$$\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)...
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
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
$$P(Y\in A | X \in A) \neq P(Y \in A) \; \; \text{for some set } A \subset \mathbb{R}.$$
...
0
votes
0answers
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?
0
votes
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 ...
2
votes
0answers
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 ...
0
votes
0answers
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 $n\rightarrow \infty$, ...
0
votes
0answers
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
$...
0
votes
0answers
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, ...
0
votes
0answers
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-...
1
vote
0answers
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.
$$\sum_{n=0}^\...
0
votes
0answers
52 views
Construction of Hermite Polynomials
I am studying Hermite Interpolation and the most common practice I came across was the use of Lagrange Polynomials. I tried to construct a Hermite Polynomial for 2 points $x_0$ and $x_1$ but instead I ...
0
votes
0answers
34 views
Prove that $\int_{-\infty}^{\infty} e^{-x^2} H_m(x)H_n(x)dx$ [duplicate]
$\int_{-\infty}^{\infty} e^{-x^2} H_m(x)H_n(x)dx=\left\{\begin{matrix}
0, & m \neq n\\
2^n n! \sqrt{\pi},& m = n
\end{matrix}\right.$
Thats a Hermite polynomials. I need a hint, i really ...
0
votes
0answers
15 views
Gaussian expectation of Lagrange polynomial
Let $\{x_i\}_{i=1}^N$ be the zeros of the probabilists' Hermite polynomial and $L_j(x)$ for $j = 1, \ldots, N$ the Lagrange polynomials with nodes $\{x_i\}_{i=1}^N$. Moreover, let $\rho \in [0, 1]$ ...
1
vote
1answer
32 views
Expanding Dawson's integral in a series of Hermite functions
We want to show that
$$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)}$$
where $F(x)$ is the Dawson Integral ($F(x)=\exp \...
3
votes
1answer
43 views
Integral Involving Hermite Polynomial and rational function
We want to prove the formula
$$\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
votes
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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)^...
3
votes
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:
$$\int_{-\infty}^{\infty}x^2e^{-x^2}H_n(x)H_m(x)dx=2^{n-1}\pi^{\frac{1}{2}}(2n+1)n!\delta_{mn}+2^n\pi^{\frac{1}{2}}(n+2)!\delta_{(n+2)m}...
1
vote
0answers
44 views
Proving that $\int_{-\infty}^{\infty}x^{m}e^{-x^2}H_n(x)dx=0$ for $m$ an integer and $0\leq m\leq n-1$
I find myself trying to prove that $\int_{-\infty}^{\infty}x^{m}e^{-x^2}H_n(x)dx=0$ for $m$ an integer and $0\leq m\leq n-1$
I tried to solve it by the following method:
Using the Rodrigues formula ...
0
votes
1answer
53 views
Spectrum and point spectrum of quantum harmonic oscillator
I have this question concerning the Hermite operator on $\mathbb{R}^n$ (defined by $Hf:=-\Delta f+|x|^2f$). We know that $H$ is a densely-defined unbounded operator on $L^2(\mathbb{R}^n)$ and that $\...
0
votes
1answer
81 views
Prove $f(\xi)$ behaves as $e^{\xi^2}$ if $\lambda<0$ in $\frac{d^2f}{d\xi^2}-2\xi\frac{df}{d\xi}+2\lambda f(\xi)=0$
How does one show that $f(\xi)$ will behave as $e^{\xi^2}$ if $\lambda<0$ in the Hermite differential equation $\frac{d^2f}{d\xi^2}-2\xi\frac{df}{d\xi}+2\lambda f(\xi)=0$
This problem comes in ...
2
votes
2answers
58 views
General solution for an integral with Hermite polynomials?
I'm currently studying quantum mechanics and thus dealing with Hermite polynomials.
By looking on Wikipedia I found this identity for Hermite polynomials:
$$\int\limits_{-\infty}^{\infty} H_m(x)H_n(x)...
0
votes
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 $$p(x_0)=f(x_0),\ p\prime(x_1)=f\prime(x_1),\ ...
0
votes
0answers
26 views
Using induction to prove a question based on Hermite polynomials [duplicate]
I am practising for a test that I have next week on proofs. I have tried to do this question using induction, but it is quite different to other induction questions I have done. Any guidance/...
1
vote
1answer
67 views
Integral representation of Hermite polynomial
I'm studying Merzbacher's Quantum Mechanics. In his treatment of harmonic oscillators, he introduces Hermite polynomials and their properties. He mentions that the Hermite polynomials (in the form ...
0
votes
0answers
49 views
I am unsure about this question using hermite polynomials and satifying recurrence relations
How would I answer a question like this? It is in a practise question for a test.
3
votes
1answer
81 views
Integral relation between Hermite and Laguerre polynomials
I'd like to proove the following integral relation
$$ \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\...
5
votes
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 ...
0
votes
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 ...
0
votes
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 $\...
0
votes
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 ...
1
vote
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 ...
1
vote
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*}
\...
2
votes
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,
$$
G(x,t) = e^{2tx - t^2}
$$...
0
votes
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?
0
votes
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 ...
11
votes
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 $n= 1$:
$$\begin{align}
xf(x) - c_1f'(x) &= 0 \\
\quad\...
1
vote
1answer
54 views
New Hermite polynomial identity? (Vol II)
I am trying to prove an identity involving Hermite polynomials using other identities from Wikipedia, but I can't find the way. I have checked the identity in Mathematica for many values of $n$ and it ...
1
vote
0answers
36 views
How to find the general solution using the Hermite polynomial?
I want to solve this equation
$${\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}\...
2
votes
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 ...
