Questions tagged [orthogonal-polynomials]
Questions pertaining to certain sets of polynomials that satisfy an orthogonality criterion with respect to some specified inner product.
573
questions
1
vote
1answer
14 views
need more explanations on the proof about zeros of orthogonal polynomials
I do not understand why should not the polynomial $P_n(x)(x-x_1)...(x-x_n)$ change sign ?
0
votes
2answers
26 views
Chebyshev in $n$ dimensions
The Chebyshev monomial integrals (of first and second kind) are
$$
I_k = \int_{-1}^1 x^k (1-x^2)^{\mp 1/2} \,dx
$$
Is anything known about their $n$-dimensional generalizations
$$
I_{k_1,\dots,k_n} = \...
1
vote
1answer
22 views
Orthogonal polynomials symmetric about centre of interval
Let $\omega(x)$ be an even weight function and $[-a,a]$ a symmetric interval about $0$. How can I prove that the orthogonal polynomials with respect to this weight function satisfy $\phi_k(-x)=(−1)^k\...
0
votes
0answers
13 views
Laguerre polynomials with integration over entire axis
The Laguerre polynomials are orthogonal with respect to the scalar product
$$
\langle f, g\rangle = \int_0^\infty f(x) g(x) \exp(-x)\,\text{d}x.
$$
Is there a class of polynomials that is orthogonal w....
0
votes
1answer
20 views
Hermite polynomials and even weight function
I looked up a problem in a book and struggled to follow the answer to it, I annotated the specific parts with (a), (b),...:
Question:
Find Polynomials $\{H_n\}_{n=0}^5$ with $H_n\in P_n$ that is ...
0
votes
1answer
31 views
Recurrence relation with derivatives
I have to show the following:
\begin{equation}
D^{n+1}e^{-x^2} = -2x D^n e^{-x^2}-2nD^{n-1}e^{-x^2}
\end{equation}
where $D^{n}=\frac{d^n}{dx^n}$
Would you give me a hint where to start?
0
votes
2answers
20 views
Finding an orthogonal basis w/ inner products
Problem
I am having trouble solving this problem. I can't find a solution and am doubting if I am right. Im thinking I let q = p2 - proj[p0, p1]p2, since that would be orthogonal to both p1 and p0. ...
1
vote
0answers
21 views
Orthogonal polynomials related to a Jacobi symmetric matrix
I would like to know if the following Hermitian tridiagonal (Jacobi) and symmetric matrix $A(t)$
\begin{equation}
A(t)=\begin{bmatrix}
0&t&0&0&0&0&0&0&0\\
t&0&1&...
3
votes
0answers
37 views
Hydrogen atom Radial wave function integrations
I wanted to do some integrations with the radial wave function for Hydrogen atom. The radial wave function is given by,
$$R_{nl}(r)=2^{l+1} e^{-\frac{r}{a n}} \sqrt{\frac{(-l+n-1)!}{a^3 n^4 (l+n)!}} \...
1
vote
1answer
60 views
A generalization of the Hermite polynomial: is there a name for this class of polynomials in the literature?
An explicit expression for the `probabilist's' Hermite polynomial is given by $$\sum_{r=0}^{\lfloor n/2\rfloor}(-1)^r\frac{n!}{2^r(n-2r)!r!}x^{n-2r}.$$ In playing around with some combinatorics, I ...
0
votes
0answers
15 views
Determinant of a Hankel matrix with gamma entries
Is there a way to show that the determinant of the matrix
$$ \begin{pmatrix} \Gamma(z+1) \Gamma(1) & \Gamma(z+2)\Gamma(2) & \cdots & \Gamma(z+n+1)\Gamma(n+1) \\
\Gamma(...
0
votes
1answer
24 views
High order (up to 4) derivatives of Chebyshev polynomials needed
I need to compute high order (up to 4) derivatives of Chebyshev polynomials at the points of the Chebyshev-Lobato grid:
$$x_j=cos(πj/N), j=0,\dots,N$$
Does anyone know how to do that? I tried ...
2
votes
0answers
41 views
Orthogonal polynomial approximation of any function
Any nonlinear function $f(x)$, $x \in \mathbb{R}^n$ can be approximated by using orthogonal polynomial approximation
\begin{equation}
f(x) = \sum_{i=0}^{N_f} b_i q_i(x),
\end{equation}
where $N_f$ ...
0
votes
0answers
22 views
Orthogonality Lagrange basis
I can't see the following:
Let $\phi_0,\phi_1,...$ be a sequence of orthonormal polynomials on an interval $[a,b]$ w.r.t positive weight function $w(x)$. Let $x_1,..., x_n$ be the $n$ zeros of $\...
0
votes
0answers
23 views
Generate Hermite polynomials orthogonal to non standard Gaussian
I am wondering how the generation of Hermite polynomials that are orthonormal to non-standard Gaussians works. I am trying to understand how to generate them because I wish to code a scaling of these ...
1
vote
1answer
32 views
Chebyshev Polynomials of the first kind
If I write the Chebyshev polynomial of the first kind like this:
$T_n(x)=\cos(n\cos^{-1}x)$ for $x$ in $[-1,1]$.
It is clear that if $x=1$ then:
$T_n(1)=\cos(n\cos^{-1}1)=\cos(0)=1$ for all $n$, ...
0
votes
0answers
41 views
Smallest root of a given polynomial when the degree tends to infinity
Frequently, in Physics, we need to determine the smallest root of a polynomial when its degree tends to infinity. Namely, let's suppose we have a polynomial of degree $d$,
$\displaystyle P_d(x) = \...
1
vote
1answer
37 views
Least square using orthogonal polynomial
I have obtained some orthogonal polynomials, using Gram orthogonal process, and the next question says, using them (O.P.) obtain the least square approximation of second degree for $f(x)=x^{3/2}$ on $[...
0
votes
0answers
19 views
Recursion relation for the expansion coefficients of the product of two Jacobi polynomials in terms of one Jacobi polynomial
I need the recursion relation satisfied by the expansion coefficients $\left\{ {{c_k}} \right\}$ in the following series:
$P_n^{(\mu ,\nu )}(x)P_m^{(\mu ,\nu )}(x) = \sum\limits_{k = \left| {n - m} \...
0
votes
0answers
45 views
Stieltjes polynomials
Is there a recursive formula to generate the Stieltjes polynomials with respect to the Legendre polynomials or the Legendre function of the 2nd kind?
Stieltjes polynomials is
$$
E_{n}(x)
$$
I've ...
1
vote
0answers
42 views
Notes on Hermite Polynomials
I am currently studying Hermite Polynomials, however every book that I have read (Mathematical Methods for Physicists-Arfken,Mathematical methods for physics and engineering-Cambridge University Press,...
0
votes
4answers
46 views
Polynomial basis - why orthogonal?
I am currently working my way through a Functional Analysis book in self-study and have come across a statement I couldn't quite follow. In one chapter, the authors present an inner product space (...
2
votes
1answer
79 views
Prove a relation of Laguerre polynomials
Prove this relation for Laguerre polynomials $L_{n}^{(\alpha)}(x)$:
$$L_{n}^{(\alpha)}(cx)=(\alpha+1)_n\sum_{k=0}^{n}\frac{c^k(1-c)^{n-k}}{(n-k)!(\alpha+1)_k}L_{k}^{(\alpha)}(x).$$
I tried to prove ...
0
votes
2answers
83 views
Find the generating function of Chebyshev polynomials
The Chebyshev polynomials of the first kind are:
$$T_n(x)=\cos(n\theta)$$ where $x=\cos(\theta)$.
Prove that the generating function of Chebyshev polynomials is:
$$\sum_{n=0}^{\infty}T_n(x)t^n=\frac{...
0
votes
0answers
20 views
Prove the orthogonality relation of Hahn polynomials
The Hahn polynomials have the representation:
$$Q_n(x)=Q_n(x,\alpha,\beta,N)={}_3F_2(-n,n+\alpha+\beta+1,-x\;; \alpha+1,-N\;;1),\;n=0,1,\cdots,N.$$
where $${}_rF_s{}(a_1,\dots,a_r;b_1,\dots,b_s;z)=\...
2
votes
3answers
59 views
Prove a relation for Chebyshev polynomials
The Chebyshev polynomials of the first kind are:
$$T_n(x)=\cos(n\theta)$$ where $x=\cos(\theta)$.
Prove relation:
$$T_n(x)=\frac{1}{2}[(x+\sqrt{x^2-1})^n+(x-\sqrt{x^2-1})^n].$$
I tried but I don't ...
0
votes
0answers
16 views
Wigner function of general Hermitte-Gaussian functions
I would like to compute the Wigner function of $\Psi_n(x)$ and $\Psi_m(x)$, with positive integers $n$ and $m$:
$$
W_{n,m}(x,\nu) = \int \! \Psi_n^*\left(x-\frac{\Delta x}{2}\right) \Psi_m\left(x+\...
1
vote
0answers
68 views
Evaluate $\int^1_{-1} \frac{P_{l_1}^{m_1}(x)P_{l_2}^{m_2}(x)}{\sqrt{1-x^2}} dx$
I wish to evaluate $\int^1_{-1} \frac{P_{l_1}^{m_1}(x)P_{l_2}^{m_2}(x)}{\sqrt{1-x^2}} dx$.
There is a neat way to show the integral is zero for certain combinations of ms and ls shown here:
...
0
votes
0answers
31 views
Methods for numerically calculating families of ON polynomials given inner product?
I am aware of some famous families of polynomials, for example Bessel, Jacobi, Chebyshev, Hermite and so on. They have one thing in common : they are all ON bases $\{P_0,P_1,\cdots\}$ with respect to ...
1
vote
1answer
40 views
How one can reformulate the sentence: $T_{n}(x)/x$ is irreducible over the integers
The motivation to this question can be found in: Chebyshev Polynomials and Primality Testing
My question is:
How one can reformulate the sentence:
$T_{n}(x)/x$ is irreducible over the integers
...
0
votes
0answers
55 views
The Bernoulli Polynomials
We know that
where $B_n(t)$ is Bernoulli polynomials.
My question: Can Bernoulli polynomials be orthogonalized with respect to a weight function $\omega$? or I mean what is a weight function under ...
0
votes
0answers
20 views
Inverse transform sampling of linear combination
Given a non-negative spherical function $f: \mathbb{S}^2 \rightarrow \mathbb{R}$:
$$f(\boldsymbol{\omega}) \geq 0, \; \boldsymbol{\omega} \in \mathbb{S}^2$$
I want to integrate $f$ over $\mathbb{S}^...
1
vote
1answer
50 views
What non-zero function is w-orthogonal to all the polynomials of degree less than or equal to $n$?
Background
It is understood that a function $q$ is $w$ orthogonal to a function $p$ over $[a,b]$ if there holds:
$$ \int_{a}^{b} q(x)w(x)p(x)dx = 0$$
For example, for $w(x):=1$, $[a,b]=[-1,1]$, the ...
0
votes
0answers
4 views
basis covering space of 2nd order spherical harmonics with no prefferential direction.
Basis of 2nd order spherical harmonics (d-orbitals or quadrupoles in physics) is typically chosen like
(https://commons.wikimedia.org/wiki/File:D_orbitals_of_an_atom.gif)
$\{ xy, yz, zx, x^2-y^2, 2z^...
1
vote
0answers
28 views
Orthogonal polynomials with weight $(1+x^{2m})^{-1}$
Very simple question. Is much known about orthogonal polynomials on the real line with weight function:
$$
w_m(x)=\frac{1}{1+x^{2m}}\exp(-\alpha\cot^{-1}(x))
$$
A reference would be much appreciated. ...
0
votes
0answers
73 views
Inner products in Gram-Schmidt orthogonalization
I am using Gram-Schmidt orthogonalization to create orthogonal polynomials with respect to an arbitrary weight function in a n-dimensional space (not always orthotope).
$P_i(\mathbf{x})=B_i(\mathbf{x}...
0
votes
1answer
32 views
Legendre polynomial : show by recurrence that $P_n(1) = 1$
Starting from the recurrent relation of Legendre polynomial :
$(n+1)P_{n+1}(x) - (2n+1)xP_n(x) + nP_{n-1}(x) = 0 $ for $n \geq 1$ with $P_0(x) = 1$ and $P_1(x) = x$
How can I show by recurrence ...
0
votes
0answers
14 views
About coefficients of truncated Fourier series
Let $f(t)\simeq a_0 + \sum_{n=1}^{r}a_n \cos (\frac{2n\pi t}{L})+\sum_{n=1}^{r}a_n^* \sin (\frac{2n\pi t}{L})=A^T \Phi (t)$ in which $A=[a_0 ,a_1 ,...,a_r,a_1^* ,...,a_r^*]^T$ and $\Phi (t)=[1,\cos (\...
0
votes
0answers
28 views
Does the following uniform estimate for the Hermite polynomials hold?
$H_n$ denotes the physicists' Hermite polynomial of degree $n$.
For $n$ large enough, I was wondering if there exists $C$ (independent of $x$) such that :
$$
e^{-x^2/4}H_n(x) < C \frac{2^{n/2}}{\...
0
votes
0answers
20 views
Why is integral a proper inner product of polynomials? [duplicate]
I am aware that it may be a duplicate question, but I spent over an hour trying to find a proper proof and I failed. One may define inner product of polynomials $p, q$ as integral: $$\langle p, q \...
0
votes
1answer
29 views
Orthogonal Polynomials approximation and $L^2(\mathbb(R))$
I have another basic question, this time about approximation of functions.
Given $(p)_{i\in \mathbb{N}}\in \mathcal{A}=\{$all the families of orthogonal polynomials $\}$ and $f\in L^2(\mathbb{R})$, ...
0
votes
0answers
15 views
Existence, derivation and properties of 2D orthogonal polynomial families?
There exist several famous families of one-dimensional polynomials, which for various different integral inner-products constructed so that every polynomial is orthogonal to every other.
Now to the ...
0
votes
1answer
76 views
non-singular Toeplitz submatrices
Let $p(x) = (1+x+x^2)^d$ for $d\ge 2$ and call its coefficients
$$
p(x) = a_0 + a_1x+ a_2x^2 + \dots + a_{2d} x^{2d}.
$$
Let $T(d)$ be the infinite upper triangular and Toeplitz matrix defined as
$$
T(...
3
votes
1answer
96 views
Prove Integral representation of Laguerre polynomials
Let $(L_n^{(\alpha)}(x))_n $ a sequence of Laguerre polynomials, for $n=0,1,..., $ and ${\alpha>-1}$, prove that :
$$ n!L_n^{(\alpha)}(x)=x^{-\frac{\alpha}{2}}\int_0^{\infty}e^{x-
y}y^{...
0
votes
0answers
30 views
Rational functions of orthogonal polynomials
Let $\Omega\in\mathbb{C}$ is defined as
\begin{equation}
\Omega = s^2\frac{LC}{2}
\end{equation}
and matrix $\mathbf{A}$ as follows
\begin{equation}
\mathbf{A} =
\begin{bmatrix}
T_{n}(1 + \Omega) ...
1
vote
0answers
35 views
I have a question on Hilbert Spaces
The question: Let $(P_n), n \geq 0$, be the Legendre Polynomials, which is a total orthonormal system in real $L^{2} [-1, 1]$ with the inner product $\langle x,y\rangle = \int_{-1}^{1} x(t)y(t) dt$. ...
3
votes
1answer
141 views
Orthogonality of Legendre polynomials from generating function
Given the the Legendre polynomials generating function:
$$G(x,t)=\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n$$
prove the relation:
$$\int_{-1}^{1} (P_n(x))^2 dx = \frac{2}{2n+1}$$
My ...
2
votes
1answer
68 views
Why does orthogonalizing the monomials give Legendre polynomials?
It's pretty well known that performing a Gram-Schmidt process on the monomials,
$$
p_j(x) = x^j - \sum_{i=0}^{j-1} \frac{\langle x^j|p_i\rangle}{\langle p_i|p_i \rangle}p_i(x),
$$
gives (scaled) ...
1
vote
0answers
31 views
For an even weight function prove that the orthogonal polynomial is even or odd function depending on the grade.
Let $w(x)$ be an even weight function and [a,b] is a symmetric region with respect to $0$. Prove that the orthogonal polynomial satisfies $p_{n}(-x)=(-1)^{n}p_{n}(x)$ for $n=0,1,2..$
It says that an ...
0
votes
0answers
80 views
Gauss Quadrature Error in 2D
Could someone tell me what is the error when applying the Gauss Quadrature rule in 2 dimensions? I know that for one dimension the error is
$$\frac{(n!)^{4}}{(2n+1)[(2n)!]^{3}} \cdot f(\xi)^{2n}(b-a)...
