Questions tagged [orthogonal-polynomials]

Questions pertaining to certain sets of polynomials that satisfy an orthogonality criterion with respect to some specified inner product.

Filter by
Sorted by
Tagged with
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)...

1
2 3 4 5
12