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

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21 views

creating a loop and maybe a potential loop within a loop in matlab [on hold]

I'm extremely new at Matlab so would like some help. I'm trying to create a loop and maybe a loop within a loop, I'm not too sure at the moment but here's what I'm trying to achieve. I began with ...
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0answers
15 views

Express Jacobi polynomial $P_n^{\alpha-1,\beta-1}$ into the sum of $P_m^{\alpha,\beta}$

I want to express Jacobi polynomial $P_n^{\alpha-1,\beta-1}$ into the sum of $P_m^{\alpha,\beta}$, where $m$ is $n,n-1,n+1...$. Can this be done? Equation (20,21) in this link might be useful.
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22 views

Polynomial Chaos with Beta Distribution - Standard Beta Random Variable, Transformation of Beta Random Variable

Background: I am dealing with a non-intrusive polynomial chaos expansion (e.g. here [Hosder,Walters;2010]). This means I want to represent an uncertain output $U(\xi)$, dependent on a vector of ...
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1answer
38 views

Finding functions orthogonal to all polynomials

I think the title is correct, but I'm not familiar with this sort of question, so maybe it's off. Anyways, the question is Find all real-valued $f$ that are continuous on $[0,1]$ and satisfy $$\...
4
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2answers
51 views

Inverting a Particular Integral Operator

Consider trying to find a function $f \in L^2(0,1)$ satisfying $$a_n = \int_0^1 f(x)x^n dx$$ Where $n$ is a nonnegative integer. Is there any method to go about doing this in general for any ...
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0answers
29 views

Combinatorial formula for Legendre Polynomials

Using the recursion formula for the solution of the Legendre equation: $$(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x) = 0$$ With solution $P_n(x)$ such that $P_n(1) = 1$, show that $$P_n(x) = \sum_{k = 0}^{n}\...
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0answers
14 views

Spectrum of Kernel - Discrete orthogonal polynomials

Trying to solve a problem, I encounter a Kernel of the form $$K(m,n)= e^{-\frac{\beta}{4} (m+n+1)} \frac{2^{2+\frac{m+n}{2}}}{\sqrt{m! n!}} \frac{\sqrt{\pi}}{n-m} \left[ \frac{1}{\Gamma(-m/2)\Gamma(...
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1answer
56 views

Prove a Hermite polynomial property by linear algebra

Let $$ X=\begin{pmatrix} 0 & & & &\\ 1 & 0 & & &\\ & 1 & 0 & &\\ & & \ddots & \ddots &\\ & & & 1 & 0 \end{pmatrix}, D=\...
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0answers
27 views

Calculating Gauss Jacobi Lobatto quadrature nodes and weights

Ke!san online calculator http://keisan.casio.com/exec/system/1329114617 has the nodes and weights quadrature calculation facility to calculate lobatto, legendre, jacobi, etc. quadratures. i want to ...
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1answer
17 views

Help computing this product through the laguerre polynomial generating function

I don't understand generating functions very well and I was told in one of my questions that the coefficient of $t^N$ in the product between $$\frac{1}{(1-t)^2}\,\exp\left(-\frac{tx}{1-t}\right)$$ and ...
3
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0answers
77 views

Two dimensional (discrete) orthogonal polynomials for regression

This question How to work out orthogonal polynomials for regression model and the answer http://math.stackexchange.com/a/354807/51020 explain how to build orthogonal polynomials for regression. ...
3
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1answer
35 views

Simplifying this series of Laguerre polynomials

I trying to figure out whether a simpler form of this series exists. $$\sum_{i=0}^{n-2}\frac{L_{i+1}(-x)-L_{i}(-x)}{i+2}\left(\sum_{k=0}^{n-2-i} L_k(x)\right)$$ $L_n(x)$ is the $n$th Laguerre ...
0
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1answer
29 views

Find an orthogonal basis for $\mathbb P_2$

The problem: For polynomials $\mathbb{P_2}$ we define the inner product between p and q as: $$ \langle p,q\rangle =p(t_0)q(t_0)+p(t_1)q(t_1)+p(t_2)q(t_2) $$ with $$t_0=0, t_1=1, \textrm{ and } ...
3
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1answer
32 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
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0answers
31 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 ...
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0answers
30 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 $...
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0answers
18 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
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1answer
35 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 ...
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0answers
19 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: $<...
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0answers
31 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
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2answers
205 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 ...
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0answers
24 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. $$ P(p_i,x)=\...
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16 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 $$ F^{-1}_X(x) = \sum_{j=0}^{N} s_j^{(X)} \psi(\xi)$$ ...
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2answers
19 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
31 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
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 f,\Phi_i\...
1
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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 ...
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1answer
40 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
52 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
12 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
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1answer
54 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
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2answers
76 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
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1answer
40 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 & \mu_{n-1}...
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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 $a$.
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3answers
28 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 polynomial ...
1
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1answer
31 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 Grahm-...
0
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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 & \...
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1answer
52 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
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0answers
41 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
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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 other,...
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0answers
33 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 $$Q_n(x)=P_n(x)\sqrt{\frac{w(x)}{r(...
0
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0answers
50 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
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0answers
49 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, $R=\sqrt{r^2+\rho^...
6
votes
1answer
151 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 2y}\...
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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
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0answers
24 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 $x=...
2
votes
1answer
34 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 $n$'...
0
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0answers
19 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
71 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
71 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 ...