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

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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 $$ F^{-1}_X(x) = \sum_{j=0}^{N} s_j^{(X)} \psi(\xi)$$ ...
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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) = ...
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0answers
24 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 ...
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1answer
21 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 ...
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0answers
13 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 ...
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1answer
24 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 ...
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2answers
48 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 ...
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0answers
9 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 $P_n$, ...
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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! ...
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2answers
62 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?
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0answers
8 views

Polynomial Chaos Representation of a lognormal RV

is it possible to represent a lognormal distributed RV via a polynomial chaos expansion (with hermite polynomials)? If not, (I guess one would need Laguerre Polynomials instead of Hermite), can ...
2
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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 & ...
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0answers
21 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 $a$.
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3answers
26 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 ...
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1answer
26 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 ...
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0answers
16 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 & ...
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1answer
38 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 ...
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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 ...
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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 ...
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0answers
28 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 ...
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0answers
26 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 ...
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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, ...
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1answer
141 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 ...
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0answers
19 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 ...
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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 ...
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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 ...
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0answers
14 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. ...
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0answers
64 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 ...
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2answers
63 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 ...
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1answer
32 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 ...
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0answers
25 views

Octahedron and System of trigonometric equations

Could somebody help me to prove the following? $$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \cos (\phi_k)=0$$ ...
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1answer
49 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? ...
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3answers
194 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 ...
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0answers
20 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 $p=2kr$ ...
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1answer
32 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 ...
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1answer
46 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 ...
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1answer
61 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: $<f(x), g(x)> = \int_{-1}^{1} f(x)g(x)dx$. ...
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1answer
73 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) = ...
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1answer
72 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 $_2F_1$: ...
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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 ...
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1answer
20 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 ...
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1answer
62 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 ...
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1answer
36 views

Orthogonality of Laguerre polynomials from generating function

I'm trying to show the orthogonality relation for Laguerre Polynomials $L_n(x)$ through their generating function $G(x,t)$. $$G(x,t)=\frac{1}{1-t}e^{\frac{-xt}{1-t}}=\sum_{n=0}^{\infty} L_n(x) t^n$$ ...
3
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1answer
46 views

Show series representation of orthogonal polynomials

wikipedia has the following series expansion for hermite polynomials, namely: $$\exp \left\{xt-\frac{t^2}{2}\right\} = \sum_{n=0}^\infty {\mathit{He}}_n(x) \frac {t^n}{n!}.$$ Does anybody see how ...
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0answers
18 views

Ortogonal polynomial regression

I need to fit a weighted set of points with orthogonal polynomials based on least squares polynomials. The grid on X is nonuniform, so Chebyshev polynomials can't be applied. The data points are like ...
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56 views

How to solve a homogeneous Fredholm integral equation of the second kind with a symmetric non-seperable kernel?

I have the equation \begin{eqnarray} \lambda L(p)=\int dq\,K(p,q)L(q) \end{eqnarray} Where $L$ is an unknown function, $\lambda$ is some constant, and $K$ is a known function. This is a homogeneous ...
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0answers
26 views

Norms of Hermite polynomials for a generalized weight function

I would like to compute the following integral $\int dz^2 e^{-a|z|^2+\frac{b}{2}(z^2+\bar z^2)}~|He_n(c z)|^2=h_n$, where $z\in\mathbb{C}$, $dz^2=dx dy$ and $He_n(cz)$ are the probabilistic Hermite ...
2
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0answers
54 views

Can anyone identify the orthogonal polynomial for this recurrence relation?

I have come across this recurrence relation: \begin{equation} x p_n(x) = (N - n)(n + 1) p_{n+1}(x) + (N - n + 1) n p_{n - 1}(x) \end{equation} with $p_{-1}(x) = p_{N + 1}(x) = 0$. I expect $p_n(x)$ ...
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1answer
25 views

Find the minimum coefficients in an inner product on L2(-1,1) using Legendre polynomials as orthonormal vectors.

Given that the orthonormal vectors in $L^{2}$(-1,1) obtained by applying the Gram-Schmidt process to 1,x,$x^{2}$ are scalar multiples of the first three Legendre polynomials: $P_{0}(x) = 1$, ...
2
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1answer
36 views

Need hints on this hermite polynomial expand

"Expand $x^{2m}$ in a series of hermite polyniomials". We start with the definition of an orthogonal expansion: $$f(x) = \sum_{0}^{\infty}\frac{<f,H_n>_wH_n}{||H_n||^2}$$ with the weight ...