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

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3
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1answer
26 views

Determinant of hankel matrix of hyperbolic functions, $a_n=\frac{n}{\sinh(\pi n)}$

I am trying to learn about the properties of Hankel matrices, and they appear to have nice closed forms for quite a large class of sequences. The class I am interested is when the elements $a_n$ are ...
1
vote
0answers
26 views

Distance in polynomial spaces

I'm trying to solve some exercises proposed by ENS of Paris. In particular the last one (which can be seen at http://www.ens.fr/IMG/file/SI2015/Sujets%20SCiences/Math2-version%20anglaise.pdf). Since ...
1
vote
0answers
25 views

Infinite Sum involving Laguerre Polynomials

I would like to simplify (if possible) $$ \sum_{k=0}^\infty(-\alpha)^k\frac{(2k)!\:L(2k,-\beta)}{k!} $$ where $L(n,x)$ is the $n$-th Laguerre polynomial evaluated at $x$. In this case, I know that ...
1
vote
0answers
13 views

finding generating function of orthogonal polynomials through their moments

I was studying a method to find the generating function of Orthogonal Polynomials through its moments. Please refer to the paper Use of Hermite's method to obtain generating functions for classical ...
0
votes
1answer
27 views

Laguerre polynomials and Gram Schmidt

Last two days I was trying to solve the following problem But I couldn't. It's a problem (#5.2.2) from Mathematical Methods for Physicists by George B. Arfken and Hans J. Weber, 7th Edition. I tried ...
0
votes
0answers
18 views

3-point recurrence relation for Hermite Polynomials

I am trying to show, given $H_{n+1} = xH_n-nH_{n-1}$, that $<H_n,H_n>$ = $n<H_{n-1},H_{n-1}>$. I have the solution available to me, from which I do not understand the following: ...
0
votes
0answers
27 views

Generating a new orthogonal base from Legendre Polynomials

I was manipulating some discrete orthogonal bases and I found this property : A complete orthogonal set of functions $\{\phi_n(x)\}_{n=0}^{\infty}$ in $\mathcal{L}^2(a,b)$ with weight function ...
5
votes
2answers
193 views

Deriving the Normalization formula for Associated Legendre functions: Stage $4$ of $4$

The question that follows is the final stage of the previous $3$ stages found here: Stage 1, Stage 2 and Stage 3 which are needed as part of a derivation of the Associated Legendre Functions ...
0
votes
0answers
23 views

What is support and spectrum of this nonnegative trigonometric function (or Finite Fourier Sum )?

This is a follow up of another question. The zeros of the following cosine sum shows the prime distribution, and the gap between the zeros can help to study the gap between prime numbers. $$ ...
0
votes
0answers
14 views

polynomial chaos expansion: linear combination properties?

I'm dealing with polynomial chaos expansion, for finite support specifically. Assume $X$ and $Y$ are r.v.'s whose the inverse CDFs expressed as $$ F^{-1}_X(x) = \sum_{j=0}^{N} s_j^{(X)} \psi(\xi)$$ ...
1
vote
2answers
18 views

Numerical algorithm: Spectral function -> Continued Fraction

I am trying to code up a numerical algorithm which takes a spectral function of the form $$c(\zeta) = w_0 +\sum_{m=1}^N \frac{w_m}{\lambda_m+\zeta}$$ into a continued fraction of the form $$c(\zeta) = ...
1
vote
0answers
28 views

L2 Norm: Unfamiliar notation

In this article that I am reading, I am given a non-negative spectral function $w(\lambda)$ which is "interpreted as a weight function determining the scalar product of two functions $f(\lambda)$ and ...
0
votes
1answer
22 views

Prove $f - p_*$ is orthogonal to all $p \in P_n[a,b]$

Let $p_* \in P_n[a,b]$ be the best $L_2$ approximation to $f \in C[a,b]$. Then $f - p_*$ is orthogonal to all $p \in P_n[a,b]$ I set: $$p_* = \sum_0^n c_i\Phi_i \text{ and } c_i = \langle ...
1
vote
0answers
14 views

Prove $\Phi_{n+1}$ is orthogonal to all p $\in P_n[a,b]$

Let $\Phi_j(j=0,1,2,...,n+1)$ be a system of orthogonal polynomials on [a,b]. Prove: $\Phi_{n+1}$ is orthogonal to all p $\in P_n[a,b]$ I'm going to solve like following: I set $p_n(x)=\sum_{k=0}^n ...
0
votes
1answer
31 views

Are linearly independent harmonic polynomials orthogonal upon integration over the sphere?

There is a theorem that states that the vector space of homogeneous polynomials decomposes into an orthogonal direct sum of vector spaces of harmonic polynomials as $$ \mathcal{P}_n = \mathcal{H}_n ...
0
votes
2answers
49 views

Understanding part of the proof that $\int_{x=-1}^1P_L(x)P_M(x)\,\rm{d} x =0\quad\text{(for}\,\, L\ne M)$

Starting from Legendre's Differential Equation $$\begin{align*} (1-x^2)y^{\prime\prime}-2xy^{\prime}+L(L+1)y=0\tag{1} \end{align*}$$ In the text that follows $P_L(x)$, $P_M(x)$ represent general ...
0
votes
0answers
10 views

How to get the coefficients between different orthogonal polynomial basis

There are two different coordinate systems. One is $ (x_1,x_2) $ and the other is $ (y_1,y_2) $ such that $ x_1 x_2=y_1, $ and $ x_2^2=y_1^2+y_2^2. $ With the following basis in Legendre $P_n$, ...
4
votes
1answer
53 views

Prove that $\int_1^{\frac{1+z}{1-z}} \frac{d^n}{dz^n} (z^2-1)^n= -\frac{2(z-1)^n}{(n+1)! }\frac{d^n}{dz^n} (\frac{z}{z-1})^{n+1} $

Prove that $$\int_1^{\frac{1+z}{1-z}} \frac{d^n}{dz^n} (z^2-1)^n= -\frac{2(z-1)^n}{(n+1)! }\frac{d^n}{dz^n} (\frac{z}{z-1})^{n+1} $$ Here are some attempts. $\frac{d^n}{dz^n} (z^2-1)^n=2^n n! ...
2
votes
2answers
70 views

Show the Integration of Legendre Polynomials is 0

Show that for the Legendre polynomial $P_n$ with $n\neq0$, $$\int_{-1}^{+1} P_n (x) dx =0$$ I put this polynomial in the Legendre equation then got stuck. Can you help me find out what to do next?
2
votes
1answer
36 views

Solution to Systems of Equations, Orthogonal Polynomials

Let $\mu_j=\int_a^b x^jw(x)dx$ be the $j$th moment of the weight distribution $w(x)$. Show that the linear system of equations $$\left[\begin{matrix} \mu_0 & \mu_1 & \cdots & ...
1
vote
0answers
22 views

Simplify Legendre Polynomial times a power

How do I simplify $x^n P_n\left(\frac{y} {x} \right) $ where $P_n$ is a Legendre polynomial and $x^2=y^2+a$? I further want to take derivatives with respect to $y$ and $a$.
-1
votes
3answers
27 views

The Meaning of the Orthogonal Projection Transformation

Let $T: \Bbb R^3 \to \Bbb R^3$ the Orthogonal Projection Transformation on the plane $x+2y+5z=0$. Find 5 non-trivial T-Invariant sub spaces of $\Bbb R^3$. I believe I have to find a ...
1
vote
1answer
28 views

Orthogonal Projection of a function

If $C[-2,2]$, let $W:= span\{x,e^x\}.$ How would I go about figuring out the orthogonal projection of $x+1$ on $W$? encountered this problem and it really has me stumped. I was told that the ...
0
votes
0answers
17 views

Generating Orthogonal polynomials for Gaussian Quadrature

I am attempting to show that if $u_j=\int_a^b w(x)x^jdx$ and $$A_n=\left( \begin{array}{cccc} u_0 & u_1 & ... & u_n \\ u_1 & u_2 & ... & u_{n+1} \\ \vdots & \vdots & ...
0
votes
1answer
44 views

Deriving Hermite polynomial derivative recurrence relation straight from differential equation.

I want to derive the derivative recurrence relation for the Hermite polynomials straight from the Hermite differential equation. That is, I want to go from left to right in the following sequence ...
0
votes
0answers
39 views

Orthogonal polynomial and inner products problem

$\{\phi_1, \phi_2,...\}$ are orthogonal polynomials with respect to some arbitrary real inner product but are normalised to be monic. The question doesnt say but I think $\phi_i$ is supposed to be of ...
0
votes
0answers
24 views

Monte Carlo integration to solve coefficients of an orthogonal series - reusing the set of random points

I'm trying to approximate a function by summing a series of orthogonal functions. $f(x) \approx \sum_i a_i \phi_i(x)$ Since the set of functions $\phi_i(x)$ are orthogonal with respect to each ...
1
vote
0answers
29 views

Derivative of Orthogonal Polynomials

Let $P_n(x)$ is a discrete orthogonal polynomial, $w(x)$ is the weight function, $r(n)$ is the norm, and $Q_n(x)$ s the weighted orthogonal polynomial such that ...
0
votes
0answers
34 views

Definite integral involving Jacobi polynomial

I need to solve the following integral: $\int_{-1}^1dx\ (1-x)^{n+2}(1+x)^{2A-k-n-1}J_{k-1}^{(2,2A+1-2k)}(x)\ \ \ \ \ \ \ $ (1) where $J_n^{(\alpha,\beta)}$ is the Jacobi polynomial. I have the ...
3
votes
0answers
48 views

Expansion of some singular kernel with the help of Bessel and Neumann spherical harmonic functions

With the following notations: $j_n$: spherical Bessel functions, $y_n$: spherical Neumann function, $P_n$: Legendre polynomial, $r$, $\rho$, $\theta$, $\lambda$ arbitrary complex, ...
6
votes
1answer
143 views

Orthogonality relation as double sum of products of binomial coefficients

I have stumbled upon the following sum over $x,y$ for non-negative integers $\kappa,\lambda$: $$ \sum_{x=0}^{\kappa}\sum_{y=0}^{\lambda}\left(-\right)^{x+y}{2\kappa \choose 2x}{2\lambda \choose ...
1
vote
0answers
22 views

Prove that the special Hermite functions are eigenfunctions of $R_{x,y}$?

How to prove that the special Hermite functions are eigenfunctions of the rotation operators $$R_{x,y} = x\frac{\partial}{\partial y}- y\frac{\partial}{\partial x}.$$ Where the special Hermite ...
1
vote
0answers
23 views

Refences to Sturm–Liouville theory with a singular weight function.

For $\alpha,\beta$, nonpositive integers at least one of which is non-zero, define $\omega\colon (-1,1)\to \mathbb{R}$ as $\omega(x) = (1-x)^\alpha(1+x)^\beta$. Then $\omega$ blows at at $x=-1$ or ...
2
votes
1answer
32 views

How can I show that this Jacobi polynomial can be expressed as the sum of these two Legendre polynomials?

Let $n\in \mathbb{N}^+$ be a positive integer. Let $L_n\colon \mathbb{R}\to \mathbb{R}$ be the $n$'th order Legendre polynomial. Let $J_n^{(\alpha,\beta)}\colon \mathbb{R} \to \mathbb{R}$ be the ...
0
votes
0answers
16 views

Expansions and Approximations of Functions

When looking at series expansions of functions, the main cases I've worked with have been Taylor series and Fourier series. Whilst the latter can arguably be viewed as a subset of the former (e.g. ...
2
votes
0answers
67 views

Inner Product of Chebyshev polynomials of the second kind with $x$ as weighting

I have tried to solve the integral $${\int_0^1 U_n (x) U_m (x)x dx },$$ where ${U_n (x) }$ denotes Chebyshev polynomial of the second kind. Solving the integration and checking the result, I ...
0
votes
2answers
68 views

Prove that the Hermite polynomials satisfy the Hermite equation.

I'm struggling with this question, which says "given the generating funcion $g(x,z)=e^{-z^2 + 2xz} = \sum_{n=0}^{\infty}H_n(x) \frac{z^n}{n!}$ prove that the Hermite polynomials satisfy the Hermite ...
1
vote
1answer
35 views

gegenbauer polynomial

Usually, Gegenbuaer polynomial is denoted by $C^{(\lambda )}_{n}(x)$ with $\lambda >-1/2$. My question: is it possible to generalize Gegenbuaer polynomial for $Re(\lambda)>-1/2, \lambda \in ...
1
vote
0answers
25 views

Octahedron and System of trigonometric equations

Could somebody help me to prove the following? $$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \cos (\phi_k)=0$$ ...
1
vote
1answer
51 views

Monomials in terms of Legendre polynomials

Is there a closed-form expression for a monomial $x^m$ in terms of a sum of Legendre polynomials $P_n(x)$? $$ x^m = \sum_n a_n P_n(x) $$ How can I determine the coefficients $a_n$ in general? ...
7
votes
3answers
206 views

Roots of the Chebyshev polynomials of the second kind.

It is known that the roots of the Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$ and they are simple (of multiplicity one). I have noticed that the roots ...
1
vote
0answers
24 views

Normalization of integral involving Laguerre polynomials

The following integral involving Laguerre polynomials came up in a quantum mechanics problem I was working on: $$\int_{0}^{\infty} | {Akp^{l}e^{-p/2} L_{n-l-1}^{2l+1}}(p)|^{2} r^2dr$$ Where $p=2kr$ ...
0
votes
1answer
35 views

from an orthonormal basis of polynomials, my corresponding coordinate matrix is not orthogonal?

how could I do it? do I just put the coordinates of each orthonormal polynomial vector into a column, w.r.t. the standard basis? for example, if my orthonormal basis is $$\{1, 2x, 3x^2\}$$ Then ...
2
votes
1answer
49 views

Calculate orthonormal basis using Gram-Schmidt

Our professor gave this exercise to help us review the topic we covered in class, but it seems my knowledge is not sufficient (or we didn't cover it in enough detail during class). Assume we are ...
3
votes
1answer
66 views

Prove any polynomial of degree n that is orthogonal to ${1, x, …, x^{n-1}}$ is a constant multiple of a Legendre Polynomial.

The Legendre Polynomials are defined by $L_n(x) = \frac{d^n}{dx^{n}} (x^2 - 1)^n$. The inner product in this case is defined on $[-1, 1]$ as follows: $<f(x), g(x)> = \int_{-1}^{1} f(x)g(x)dx$. ...
1
vote
1answer
77 views

Gram-Schmidt procedure to construct an orthogonal basis for best approximation of a function

I'm using the Gram-Schmidt procedure to construct an orthogonal basis of the $Span(1, x, x^3)$ for the interval [−1, 1]. I'm trying to determine the best approximation to $f(x) = x^5$. $p(x) = ...
1
vote
1answer
74 views

Relation between these expressions involving the Hypergeometric function and the Gegenbauer polynomials

I would like to find the relation between the solutions of a differential equation obtained by two different authors. The first solution is given in terms of the hypergeometric function $_2F_1$: ...
0
votes
0answers
16 views

Zernike Polynomials in Higher Dimensions

Let $n,l$ be integers with $n-l\geq 0$. Set $\alpha =\frac{n-l}{2}$ and $\beta =\frac{n+l}{2}.\ $Then the radial part of the Zernike polynomials in dimension 4 is given by $\tag 1R_{n}^{(l)}(\rho ...
0
votes
1answer
21 views

The existence of Gaussian quadrature weights

In the Gaussian Quadrature, $x_1,\dots x_n$ are the roots of $p_n$, where $p_n$ is a Legendre polynomial, as usual. With this, why do there exist unique $A_1,\dots A_n$ such that ...
3
votes
1answer
69 views

Symmetry planes in spherical harmonic basis

Suppose I have a function $f(x):S^2\rightarrow\mathbb{C}$ in the degree four spherical harmonic basis: $$f(\theta,\varphi):=\sum_{k=-4}^4a_kY_4^k(\theta,\varphi).$$ I have two related questions: Is ...