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Questions tagged [hermite-polynomials]

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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 ...
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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}...
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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,\...
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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} \...
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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)^{...
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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 ...
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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: ...
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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}^...
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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 ...
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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^{\...
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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)|...
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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(...
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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 ...
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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 ...
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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-...
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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)=\...
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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 $x_0=0$,...
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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]$...
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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 ...
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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 ...
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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{...
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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 ...
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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??
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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}...
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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 ...
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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 ...
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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}{\...
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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\} \...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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
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1answer
119 views

Compute the $n$th stochastic integral of Brownian motions

Show that the following equivalence of the $n$th stochastic integral $$n!\int\dots\int_{0\leq s_1\dots\leq s_n\leq t}dW_{s_1}\dots dW_{s_n}=t^{n/2}H_n \left(\frac{W_t}{\sqrt{t}} \right)$$ ...
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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
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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: $$ \...
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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 ...
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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 ...