Questions tagged [hermite-polynomials]
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102
questions
0
votes
0answers
21 views
Approximation to the $n$-th derivative using reproducing kernels.
For integrable functions defined on the real line, the normalized gaussian function approximates the convolution identity, Dirac Delta, in the sense that if
$$g(t):=N_0e^{-x²}$$
(denoting the ...
1
vote
1answer
22 views
Can $\frac{d^k}{dx^k} e^{\frac{x^2}{2}}$ be written in terms of Hermite polynomials?
We know that $\frac{d^k}{dx^k} e^{-\frac{x^2}{2}}$ can be written in terms of Hermite polynomials as
\begin{align}
\frac{d^k}{x^k} e^{-\frac{x^2}{2}}= (-1)^k e^{-\frac{x^2}{2}} H_{e_k}(x)
\end{align}...
1
vote
0answers
43 views
Can a generalization of the Hermite function-like “basis” $φ_n(x) = D^n \exp(-x^2/2)$ be orthonormal in $L^2$?
I'm trying to find an orthonormal basis with respect to a weight function $w(x) \equiv 1$
for $L^2$ in terms of the following proposed basis functions:
$$ φ_n(x) = D^n \exp(-x^2/2),\quad n=0,1,2,\...
2
votes
1answer
74 views
Function series involving Hermite Polynomials
I am wondering if there is a simpler form of the following function series involving the (physicists') even Hermite polynomials:
\begin{equation}
f(x) = e^{-\frac{x^2}{2}} \cdot \sum_{n = 0}^{\infty} \...
1
vote
1answer
30 views
Even and odd Hermite polynomials
I have to derive the following relation: $$H_{2n}(0)= (-1)^{n} \frac{(2n)! } {n! } $$ where $H_{n} (x) $ are Hermite polynomials. What I've done so far:
1) tried using this formula: : $H_{n}(x)= (-1)^{...
0
votes
1answer
12 views
How to create a Cubic Hermite Spline interpolation equation?
I am developing a program for which I need to smoothly interpolate between some control points.
Here is an example of what I need. Ie. In this gif, the knob is crossfading between no interpolation at ...
0
votes
1answer
27 views
Quantum Harmonic Oscillator via Separation of Variables and/or Normalization
I tried to solve the problem of the Quantum Harmonic Oscillator in one dimension.
$$\frac{-\hbar ^2}{2m} \Psi _{xx} + \frac{1}{2}m(\omega x)^2\Psi = i\hbar \Psi _t$$
I set up the following condition: ...
1
vote
0answers
19 views
Normalized probabilist's Hermite polynomial sum
The normalized probabilist's Hermite polynomials are $\frac{1}{\sqrt{n!}}He_n(x)$, and they satisfy orthonormality. Are there any simple formulas for the following two sums,
$$\displaystyle\sum_{n=0}^...
1
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0answers
52 views
Inner product of Hermite polynomial
How to prove the Fourier Hermite series $$\int H_n(x) f(x) \phi(x) dx=\int f^{(n)}(x) \phi(x) dx $$
where $\phi(x)=\frac{1}{\sqrt{2\pi}} e^{-x^2/2}$, $f^{(n)}(x)$ is the $n$th derivative of the ...
0
votes
1answer
29 views
Derivate of $e^{-x\cdot x}$ with Hermite polynomials.
Define the Hermite polynomial $H_n$ by
$$H_n=(-1)^n e^{x^2}\frac{d^2 e^{-x^2}}{dx^n}.$$
Now let $x\in\mathbb{R}^d$, $\alpha\in\mathbb{N}_0^d$ be a multi-index and
$\partial^\alpha=\frac{\partial^{\...
1
vote
0answers
25 views
Relation on maxima of some Hermite function
Define the function :
$$
x \to z_n(x)=\frac{e^{-x^2/2}He_n(x)}{\sqrt{(n+1)!}}
$$
where $He_n$ is the probabilistic Hermite polynomials
I'd like to show (see figure below) that : $max_{x\geq0} |z_n(x)|...
0
votes
0answers
23 views
Hermite interpolation with divided differences
the following facts are given
$x_0 = 0 \;\;\; x_1 = 1$
$f(x_0)= 2 \;\;\; f'(x_0)=0$
$f'(x_1) = 0 \;\;\; f''(x_0)=0$
$f''(x_1) = 0 \;\;\; f[x_0,x_0,x_1,x_1] = 1$
It is now asked to calculate $f(...
0
votes
1answer
47 views
Integration of Hermite polynomials
Is there any closed-form expression for the integral
\begin{equation}
I(n,m) = \int_0^{+\infty}
\mathcal{H}_n(u) \mathcal{H}_m(u)\exp(-u^2) du
\end{equation}
where $\mathcal{H}_n(x)$ is the Hermite ...
1
vote
1answer
27 views
Error Estimate of Hermite Cubic Piece-Wise Interpolation
Assume $s_i$ is the Hermite cubic polynomial interpolation on $[x_i,x_{i+1}]$ such that $s_i(x_i)=f(x_i), s_i(x_{i+1})=f(x_{i+1}), s_i'(x_i)=f'(x_i), s_i'(x_{i+1})=f'(x_{i+1}), i=0,...,n-1$.
For ...
1
vote
1answer
16 views
Hermite functions: Global maxima form decreasing sequence?
Let
\begin{equation}
\psi_n(x) = \left(\frac{1}{\pi} \right)^{\frac{1}{4}}\frac{1}{\sqrt{2^n n!}} H_n(x) e^{- \frac{x^2}{2}}
\end{equation}
be the n-th Hermite function, where $H_n(x)$ denotes the "n-...
0
votes
0answers
49 views
Projection on Hermite basis with Gaussian measure different formulations
I want to do a projection of a function on a Hermite basis.
Do to so, we introduce the Hermite basis. The Hermite polynomials are constructed from the derivation of the Gauss measure.
Thus,
$$w(x)=\...
1
vote
1answer
56 views
cubic Hermite interpolation
Professor gave us this little bastard of a question and I'm at a complete loss about what to do. Some help or hints would be immensely appreciated, translated to the best of my abilities.
Let $x_0=0$,...
0
votes
0answers
42 views
Get approximation of integral using spline and simpson rule
I would like some help figuring out how to do this:
I was given a function - $f$ and $f'$ With 7 consecutive derivatives in [a,b]
,samples: $x_k = a+kh$ $h=\frac{b-a}{n}$ ,
a spline function S:[a,b]$...
1
vote
0answers
48 views
2D Hermite interpolation on rirregular points
I am trying to generalize the principle of Hermite interpolation to 2D, over irregular data points.
In 1D, we have a sequence of points on which the value of a function and its first derivative is ...
0
votes
2answers
30 views
For $y''-2\alpha y'+\frac{2}{x} y=0$ , provided that $y(x)=\sum_{n=0}^{\infty} a_nx^n$ show you can determine $a_{n+1}$ from $a_n$
Given the differential equation $y''-2\alpha y'+\frac{2}{x} y=0$,
show that if $y(x)=\sum_{n=0}^{\infty} a_nx^n$ is a solution of the
differential equation, then $a_0=0$ and that we can ...
2
votes
1answer
191 views
Finding an upper bound for a Hermite Polynomial
Here is the problem:
Let $H_5(x)$ be the Hermite polynomial that interpolates $f(x) = \text{sin}x$ at the points $-1, 0, 1$. If $\int_{-1}^{1} H_5(x)dx$ is used to approximate $\int_{-1}^{1} \text{...
0
votes
0answers
50 views
Hermite polynomial
\begin{equation}h''(x) - 2xh'(x) = -2 \lambda h(x), x \in R \end{equation}
where $\lambda$ is the eigenvalue.
\begin{equation} G(t,x) = \sum_{n=0}^ {\infty} t^nh_n = e^{2tx-t^2} \end{equation}
This ...
1
vote
1answer
29 views
Boundedness of the Hermite functions.
Is it true that Hermite’s functions are bounded for any $n\in\mathbb{N_0},x\in \mathbb{R}?$ If it's true, how to prove it??
3
votes
0answers
293 views
Generation of Hermite polynomials with Gram-Schmidt procedure
I want to use the Gram-Schmidt procedure to generate the first three Hermite polynomials. Given the set of linearly independent vectors $\{1,x,x^2,...\}$ in the Hilbert space $L^2(R,e^{-x^2}dx)$, I ...
0
votes
0answers
95 views
Integral of two Hermite polynomials, one shifted, and exponentials
I have to calculate the integral
$I=\int_{-\infty}^{\infty} H_{m}(x) e^{-x^{2}/2} H_{n}(x-a) e^{\frac{(x-a)^{2}}{2}}dx$
where $H_{n}(x)$ are the physicist's Hermite polynomials. This integral exists ...
0
votes
0answers
26 views
How do I expand the Hermite Cubic Spline basis to the nth order?
A useful basis for cubic polynomials are those used for Hermite interpolation:
$$h_{00}(t) = 2t^3-3t^2+1$$
$$h_{10}(t) = t^3-2t^2+t$$
$$h_{01}(t) = -2t^3+3t^2$$
$$h_{11}(t) = t^3-t^2$$
It is also ...
1
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0answers
25 views
Integrating expression containing a derivative
Is there a method available to obtain a closed-form solution to an integral in the following form:
$$I(x)=\int_{\mathcal{A}} \mathrm{d}x f(x)\left(\frac{\mathrm{d}^n}{\mathrm{d}x^n}g(x)\right)$$
in ...
1
vote
1answer
251 views
Hermite interpolation code in MATLAB
This is my MATLAB code for divided differences and Hermite interpolation, but it doesn't work properly. Could you take a look at it? Thank you.
I'm sorry for the layout, but it's the best I could do.
...
0
votes
0answers
47 views
Integral which involves two Hermit polynomials (Quantum forced harmonic oscillator)
I'm trying to understand the derivation of a formula which gives the probability of a quantum forced harmonic oscillator to transit to the state $n$, at instant $t$, if it was at state $m$, at instant ...
0
votes
0answers
27 views
Integral of product of Hermite functions over finite interval
I am working with the Hermite functions $h_n(x)$ such that $\int_{\mathbb{R}}h_n(x)h_m(x)e^{-x^2/2}dx=\delta_{n,m}.$ So, if $m\neq n$, we know that the integral $\int_{\mathbb{R}}h_n(x)h_m(x)e^{-x^2/2}...
-1
votes
1answer
128 views
Weber-Hermite differential equation
I was solving a quantum mechanics problem (harmonic oscillateur) and i need to solve this Weber-Hermite differential equation in an analytic method:
$$y"-x^2(y)=0$$
I know the solution of this ...
2
votes
0answers
70 views
Calculating Hermite Expansion Coefficents of $|x|$
I'm struggling to calculate the coefficents for the Hermite Expansion of the absolute value function and the indicator function $x \mapsto \mathbb{1}_{|x-u|\leq \delta}$
Background:
I know, that for ...
3
votes
0answers
72 views
$L_2$ scalar product between Hermite polynomials
I am trying to compute the $L_2$ scalar product between (probabilists’) Hermite polynomials (defined as in Wiki) with Gaussian weight and different scales, i.e. for some constants $c, d$:
$$\frac{1}{\...
6
votes
3answers
193 views
Weird Integral Involving Hermite Polynomials
I have stumbled upon the following integral involving the Hermite polynomials:
$$ I(m) = \int_\mathbb{R} e^{i m x} \left[ e^{-\frac{x^2}{2}} H_m(x) \right] dx \, , \quad m \in \mathbb{N} \cup \{0\} \...
3
votes
1answer
141 views
An Addition formula for Hermite polynomials
My question concerns an addition formula that can be found on the Wikipedia page of Hermite Polynomials but I can't find it anywhere else. The well-known formula that can be found in many books is the ...
2
votes
1answer
60 views
Sum involving Hermite polynomials
I am wondering if there is a simpler form of the following summation involving the (physicists') Hermite polynomials:
$$\sum_{k=0}^{n}\frac{H_k(x)}{(2i)^k},$$
where $i=\sqrt{-1}$ is the imaginary ...
3
votes
1answer
142 views
Integral of Hermite Polynomials.
Let $f \in L^2(\mathbb{R}, (1/\sqrt{2 \pi})\exp(-x^2/2))$ be such that $0 \leq f(x) \leq 1$.
We know that the (normalized) Hermite polynomials are a complete orthonormal basis for this space. ...
1
vote
0answers
92 views
How to prove that the hermite polynomials are a Appell sequence?
The hermite polynomials I am saying is defined by the recursive relation $f_{n+1}(x)=2xf_n(x)-f_n'(x)$ with $f_0=1$. I need to prove that they have the relation $f'_n(x)=2nf_{n-1}(x)$. I tried ...
4
votes
0answers
81 views
Properties of the differential entropy of Hermite functions
Preliminaries: let $f_n(x)\,, n=0,\ldots,\infty$ denote the Hermite functions, which are of the form
$$ f_n (x)=\#\, e^{-\frac{x^2}{2}} H_n(x) \,,$$
where $H_n(x)$ are the physicists' Hermite ...
1
vote
1answer
42 views
Seem to be able to construct function on the whole real line from the value in a finite interval. What is wrong with the argument?
I have a question I am phrasing in terms of expansion of a function in terms of Hermite polynomials but it applies to other expansions as well.
First I establish my convention for Hermite polynomials ...
0
votes
3answers
89 views
Integrating products of Hermite polynomials
Given Hermite polynomials $\Phi_0 = 1, \Phi_1 = \xi, \Phi_2 = \xi^2 - 1, \ldots$, I want to calculate
$$
\int_{-\infty}^\infty W(\xi) \prod_{n \in N} \Phi_n(\xi) \:\mathrm{d} \xi
$$
where $N$ is any ...
0
votes
0answers
40 views
Using hermite interpolation to calculate tan$(\pi x)$
I am asked to calculate tan$(\pi x)$ using Hermite interpolation in $0$ and $45$ degrees. Then bound the error made and compare the bound found with the exact error.
I am not sure if I understand the ...
0
votes
1answer
40 views
Moving around operators under integral
When trying to find the normalization constant of
$$\int_{-\infty}^{\infty}e^{-x^2}H_m(x)H_n(x) \,dx$$
for the Hermite polynomials $$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2},$$
my reference ...
6
votes
1answer
2k views
Deriving Rodrigues Formula and Generating function of Hermite Polynomial from $H_n(x)= e^{x^2/2}(x-\frac{d}{dx})^ne^{-x^2/2}$
There are a variety of ways of first defining the Hermite Polynomials in a certain way and then to derive alternative representations of them. For example in Mary Boas' Mathemmatical methods (p. 607, ...
1
vote
0answers
35 views
Do the known Hermite constants hold for complex-valued lattices?
I understand that there are some Hermite constants that have been known, e.g., Hermite constant for lattices of dimension 2, which is $\sqrt{4/3}$. But these constants are for real-valued lattices. Do ...
2
votes
1answer
119 views
Compute the $n$th stochastic integral of Brownian motions
Show that the following equivalence of the $n$th stochastic integral
$$n!\int\dots\int_{0\leq s_1\dots\leq s_n\leq t}dW_{s_1}\dots dW_{s_n}=t^{n/2}H_n \left(\frac{W_t}{\sqrt{t}} \right)$$
...
0
votes
0answers
82 views
Hermite Polynomial Expansion
Suppose I have expanded the function $f(x)$ in Hermite polynomials,
\begin{equation}
a_n = \int_{-\infty}^\infty f(x)\exp(-x^2)\mathcal{H}_n(x)dx \,.
\end{equation}
How can I express the expansion ...
3
votes
4answers
166 views
A polynomial parametric curve spanning known tangent end-points
Let $x(t)$ and $y(t)$ be unknown polynomials (of maximum order 3) defining a parametric curve
$$ \mathbf{r}(t)=\begin{bmatrix}x(t)\\y(t)\end{bmatrix} $$
that fits known tangential end-points:
$$ \...
0
votes
1answer
145 views
Hermite polynomials $H_{n}(y)=\frac{1}{\sqrt{2^n}}\left( y -\frac{d}{dy} \right)^n$ equivalent form
I want to transform the following espression:
$$H_{n}(y)=\frac{1}{\sqrt{2^n}}\left( y -\frac{d}{dy} \right)^n$$
in
$$H_{n}(y)=(-1)^n e^{y^{2}}\frac{d^n}{dy^n}e^{-y^{2}}$$
Honestly, I have no idea how ...
1
vote
0answers
207 views
Multivariate normal/change of variables in integral (“derivative of change is change of derivative”?)
Let us consider the following integral:
$$\int_{\mathbb{R}^d} |D^{\alpha} P(x;\Sigma)| dx,$$
where $P(x;\Sigma)=((2\pi)^d |\Sigma|)^{-1/2}\exp(-\frac{1}{2}x^T\Sigma^{-1}x)$ denotes the density of a ...
