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

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2
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56 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
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2answers
54 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
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1answer
29 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
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0answers
22 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
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1answer
43 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
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3answers
162 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
13 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
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1answer
23 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
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1answer
37 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
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1answer
50 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
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1answer
54 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) = ...
2
votes
1answer
70 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
14 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
15 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
56 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 ...
3
votes
1answer
22 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
votes
1answer
45 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 ...
0
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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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0answers
44 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
24 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
52 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 $p_n(x)$ ...
0
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1answer
17 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
votes
1answer
34 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 ...
0
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0answers
27 views

Does the following integrand have an evaluation:

Does the following integrand have an evaluation: $$ \int^\infty_0 f(y)\: y^{-{1\over2}}\exp\left(-y\right) L_n^{\left(-{1\over2}\right)}(y) dy $$ Where, $f(y) = \exp\left(-\alpha y^{1.5}-\beta ...
0
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0answers
6 views

Determine multipole coefficients for an abitrary function

I have an arbitrary function $\omega(r,\theta)$ which I am trying to decompose into multipole moments. As I understand, the cylindrical exterior multipole expansion is an orthogonal basis and is given ...
0
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1answer
45 views

prove two functions are orthogonal

I am new to this topic. Generally, what is the process of proving two polynomials orthogonal to each other on some given interval? Feeding some input and see that inner products are zero can probably ...
2
votes
1answer
42 views

power series expansion of the Confluent Hypergeometric Function

What is the power series expansion of the Confluent Hypergeometric Function of the Second Kind given by $U(a,b,x)$ ? what is the derivative of this function with respect to x.
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0answers
38 views

Power series expansion of associated legendre polynomial.

What is the power series expansion of the associated legendre polynomial given by $P_l^m(z)$ ? what is the derivative of this function with respect to z.?
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0answers
23 views

Build polynomial orthogonal to set of other pre-defined polynomials

Basically. the question is simple. Is there any algorithm so I can build a polynomial, orthogonal for the set of pre-defined polynomials? I need the algorithm(like Gram-Schmidt) that would be ...
0
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0answers
24 views

How to orthogonalize piecewise polynomial with respect to set of piecewise polynomials?

Currently I'm working with multiwavelet basis(need it for solving the stochastic ODE system as described here) and can't understand how to build it. Basically, I have a set of functions $p_{i} = ...
1
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1answer
49 views

How to find the weight function, with respect to which a family of polynomials is orthogonal?

When iteratively findind the best polynomial approximation wrt. $\left\|\cdot\right\|_{\infty}$, a good starting point for the process is the projection onto the basis of Chebyshev 1st kind ...
2
votes
2answers
50 views

Integral with Chebyshev polynomials

As much as I try, I can't seem to find in any book or paper how we obtain the error of the Gauss-Chebyshev quadrature formula of the first kind. I found only that the error is given by $$ ...
0
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0answers
78 views

How to find the three term recurrence relation for orthogonal polynomials given the corresponding eigensystem?

I'm interested in the Gaussian quadrature rule that uses abscissas $\{x_1, x_2, ..., x_{n - 1}, 1\}$ and weights $\{w_1, w_2, ..., w_n\}$ to approximate $\int_{-1}^1f(x)dx$ such that the approximation ...
0
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1answer
48 views

Definite Integals Obtained From Approximations Using Chebyshev Polynomial

Was trying to use chebyshev polynomial to obtain a cubic approximation to $f(x)=\frac{1}{x} $ I did it over the interval $[-1,1] $ Solving gave me the following four definite integrals: ...
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0answers
30 views

“Second kind” orthogonal polynomials and functions

Recently I've been doing reading in the subject of orthogonal polynomials on the real line (OPRL). Such OPs arise in solving the three-term recurrence relation $$x ...
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1answer
43 views

$L^2$ product of Chebyshev polynomials and Legendre polynomials

The following was a problem in a recent numerical analysis exam: Let $k \in \mathbb{N}\setminus\{0\}$. Prove or disprove: $$ \int_{-1}^{1} cos\left(k \operatorname{arccos}(x)\right) \cdot ...
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1answer
22 views

Orthogonality of Jacobi polynomials and the parameters

Consider he Jacobi polynomials denoted by $ P_n^{(\alpha,\beta)}(x) $ corresponding to the weight function $w(x) = (1+x)^{\alpha } (1-x)^ {\beta} $ defined on $(-1,1)$. Why is that the condition $ ...
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0answers
36 views

Does the following integrand have an evaluation?

One of the many terms in an equation I have derived has the following: $\int^{+\infty}_{-\infty} \text{sin}^2\left[a(1+erf(x))\right]H_n(x)\text{exp}(-x^2)dx$ H is the Hermite polynomial and n is an ...
1
vote
2answers
71 views

orthogonal polynomials in (-1,1) with a modified weight function

I would like to find the orthogonal polynomial system $ \{ P_n(x), n \in \textbf{Z} \} $ corresponding to the weight function $ w(x) = \frac { 1} {\sqrt{1-x^2} (1+\sqrt{1-x^2} ) } $ defined on the ...
1
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1answer
45 views

Simple zero of complex orthogonal polynomials

Let $\mu$ a positive measure on $\Omega\subset\mathbb{C}$ and we assume that all moment of $\mu$ exists (i.e) for all $i,j\in \mathbb{N}$ $$ \int_\Omega z^i\bar{z}^j d\mu(z)\in\mathbb{C} $$ So on ...
0
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1answer
26 views

integration involving Hermit function

Iam trying to evaluate the integration of the following two products .. every one by itself where Hn-1 is the Hermite function . All my tries ends with a zero value for the two integrations, but it ...
-1
votes
1answer
48 views

How to apply Zernike polynomials for circles with radius bigger than one?

In optics usually Zernike polynomials are used to represent the aberration of a lens. As you know Zernike polynomials are defined in the unit disk and there they are orthogonal. The problem is that ...
0
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0answers
34 views

Convergence of orthogonal basis functions

I'm working on a problem where I've generated a set of basis functions using a Laplace series of spherical harmonics to describe the angular part of a 3D distribution and "custom made" basis functions ...
0
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1answer
81 views

Cubic function: Cardano's method

(Wikipedia link) So I am writing an essay on different ways to solve cubics. But I get stuck in the Cardano's method... Mainly is the part with Cardano's method's condition $\frac{q^2}{4} + ...
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1answer
45 views

Prove that $T_n$ satisfy $ \sum_{k=0}^{N-1}{T_i(x_k)T_j(x_k)} = \begin{cases} 0 &: i\ne j \\ l\neq 0 &: i=j \end{cases} \,\! $

The Chebyshev polynomials of the first kind satisfy the recurrence relation $$ \begin{cases} T_{n}(x)=2xT_{n-1}(x)-T_{n-2}(x) \qquad n \geq 2 \\ T_{0}(x)=1, \ \ T_{1}(x)=x \\ \end{cases} $$ The ...
2
votes
2answers
49 views

orthogonal polynomials and weight functions

how are orthogonal polynomials related to their weight function? is there an algebraic relationship other than the defining integral $$\int_a^b w(x)P_n(x)P_m(x)\,dx$$? thanks for the help!
0
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1answer
78 views

Orthonormal Basis of a function

An Orthonormal Family $\{e_k\}_{k\in\mathbb{N}}$ is a basis if and only if $$f=\sum^\infty_{n=1}\hat{f}(n)e_n \ \ \ \text{in} \ \mathcal{L}^2(\mathbb{R})$$ where $f\in\mathcal{L}^2(\mathbb{R})$ ...
1
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1answer
40 views

Why an orthonormal polynomial set over a continuous domain is not over a discrete one?

I would like to read the proof showing that a orthonormal polynomial set over a continuous domain is neither orthonormal nor complete over a discrete values on that domain. For example, Zernike ...
1
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1answer
69 views

Expressing associated Legendre polynomials in terms of unassociated Legendre polynomials

The associate Legendre equation is given as: $(1-x^2)\frac{d^2}{dx^2}y-2x\frac{d}{dx}y+\left[n(n+1)-\frac{m^2}{1-x^2}\right] y=0$ This becomes the standard unassociated Legendre equation for $m=0$. ...
1
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1answer
31 views

orthogonality of hermite polynomials when $x \rightarrow f(x)$

We know that if $m \ne n$: $$ \int_{-\infty}^{\infty} e^{-x^2}H_n(x)H_m(x)dx = 0 $$ And $$ \int_{-\infty}^{\infty} e^{-x^2} H_n(x)^2dx = \delta_{nm} 2^n n! \sqrt{\pi} $$ Now I am trying to find ...