Questions tagged [orthogonal-polynomials]

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

Proving that Legendre polynomials decreasing about $n$ as $x$ approaches to 1.

I am in the study of the Legendre polynomials and think about proving $$P_n(x)>P_{n+1}(x)$$ when $x$ is very close to 1. This is obvious when I see this picture on Wikipedia. I was wonder if there ...
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48 views

Basic Proof of Properties of Kravchuk Polynomials

Fix $n$ and consider the family of polynomials $$ K_s(\ell) = \sum_{k=0}^s(-1)^k\binom{\ell}{k}\binom{n-\ell}{s-k} \quad \text{defined for} \quad 0 \leq s \leq n $$ They are called Kravchuk (or ...
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How to prove spherical harmonic addition theorem

I have been trying to prove that $$ P_\ell(\cos\gamma)=\frac{4\pi}{2\ell+1}\sum_{m=-\ell}^\ell Y_{\ell m}^{*}(\theta', \phi')Y_{\ell m}(\theta, \phi) $$ for $\cos\gamma=\cos\theta\cos\theta'+\sin\...
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29 views

Orthogonal polynomial and Taylor series approximation

For an analytic nonlinear function, does the statement always hold ``Orthogonal polynomial based linearization (first-order approximation) will be expected to be more accurate than first-order Taylor ...
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1answer
33 views

Orthogonal polynomials equations

Let ${Q_n(X)}_{n \in N_0}$ sequence of orthogonal polynomials with respect to the weight function $p(x)$ on interval $(a,b)$ . Let $x_i$, $1 \leq i \leq n$ be zeros of $Q_n$. How to prove that $$ \...
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Question regarding positivity for sequence of sums of integrals of Laguerrre polynomials.

Inspired by this question of a sequence of weighted sums of Laguerre polynomials with alternating signs being positive for $x>0$. We showed that proving that the weighted sum has no zeros for $x>...
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Is the sum of the first N Laguerre polynomials (with alternating signs) always positive?

I have noticed that the following simple sum of Laguerre polynomials (weighted with alternating signs) seems to be positive for any $N$ when $x>0$: $$\sum_{k=0}^{N}\;(-1)^{k}\;L_{k}(x)$$ More ...
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Orthogonal space to a subspace of L2

Consider $L^2([-1,1])$ and $$ V= \{f(x) = ax + bx^2 : x \in [-1,1], a,b \in \mathbb{R}\}$$ Find $V^\perp \subset L^2([-1,1])$. I'm having issues answering this question. Is there a way to find $f(x)$ ...
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$P=\sum_{k=0}^{n}c_{k}z^{k}$ if $\forall t\in[k].c_t\in\mathbb{R}\rightarrow\int_{-1}^{1}P^2\leq\pi\sum_{k=0}^{n}c_{k}^{2}$ [duplicate]

$P=\sum_{k=0}^{n}c_{k}z^{k}$ if $\forall t\in[k].c_t\in\mathbb{R}\rightarrow\int_{-1}^{1}P^2\leq\pi\sum_{k=0}^{n}c_{k}^{2}$ My try: it's clear that $\frac{1}{2\pi}\int_{0}^{2\pi}|P(e^{i\theta})|^{2}d\...
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47 views

Prove $\int_0^\infty e^{-x} x^k [L^k_n (x)]^2 \, dx=\frac{(n+k)!}{n!}$

How can I prove the normalization ratio of associated Laguerre polynomials: $$\int_0^\infty e^{-x} x^k [L^k_n (x)]^2 \, dx=\frac{(n+k)!}{n!}$$ using the generator function of Laguerre polynomial, $$\...
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Showing the following inequality by using orthogonal polynomials with even weight function Is my solution is right? how do I procced from here?

Let $\rho(x)$ be an even weight function on a symmetric interval $[−b, b]$, and let $\left\{\psi_{n}\right\}_{n=1}^{\infty}$ be normalized orthogonal polynomials. Let $f(x)$ be a piecewise continuous ...
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Minmax approximation using Chebyshev polynomial

Find minmax approximation to f(x)=|x| in $P_3$ on [-1,1] .What is the minmax error I do not know how to do that ,i got hint that i have to use Chebyshev polynomial for approximating |x|
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Basis for orthogonal complement given inner product

Consider $\langle f,g\rangle = \int_0^{\infty} f(t)\overline{g(t)}e^{-t}dt$ over the space $\mathcal{P}(\mathbb{R})$ of complex-valued polynomials in one variable. I have already proved that this map ...
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Why are these polynomials built from Jacobi polynomials orthogonal?

Let $P_i^{(a,b)}$ denote the Jacobi polynomials. I stumbled on this conjecture playing around with mathematica. As far as I can tell, for all $i,j$, natural numbers we have $$ \int_{-1}^1 \Big(\frac{\...
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A good practise book suggestion

I am a beginner in Special Functions. So I am looking for a Good Practise/Application Book about the properties of Hypergoemetric Function and its relations with orthogonal polynomials. I want to ...
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Expand Associated Legendre Polynomials in the basis of shifted Associated Legendre Polynomials

I'm tasked to solve the following integral: $$\int_0^{\pi/2}P^m_n(cos(\theta))P^m_l(cos(\theta))sin(\theta)d\theta$$ where $n,k\in Z^+$ and $m \in Z^{0+}$. Is it possible and wise to perform the ...
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Find a polynomial $P$ of the smallest degree such that $\int_{-1}^{1}\left|x^{2 / 3}-P(x)\right|^{2} d x<0.01$

Find a polynomial P of the smallest degree such that $$\int_{-1}^{1}\left|x^{2 / 3}-P(x)\right|^{2} d x<0.01$$ I am ususally solve this kind of "best approximation" problems by taking ...
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Find the range and null space of T.

The inner product space with $\langle f(x),g(x)\rangle=\int_0^2 f(t)g(t)dt$ on $P_1(\mathbb{R})$. I need to prove that $T(f(x))=f(x)-f(1)$ is an orthogonal projection. I know that I need to show $R(T)^...
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Derive the Rodrigues formula for orthogonal polynomials in general

In our math lecture we started to deal with "orthogonal polynomials". I have to say that I don't understand some things. Therefore, I did a lot of research on orthogonal polynomials and ...
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Zernike Polynomial Coefficient Calculation

Any 3D data can be expressed in cartesian co-ordinates using zernike polynomials with the following formula The values for i in the formula is from 1 to 51. There is a formula for calculating ...
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To reduce quardatic form to cononical form by orthogonal method

$2(x_1)^2+2(x_2)^2+2(x_3)^2+2x_1x_3$ Image for original question --> https://i.stack.imgur.com/Qmb2i.jpg
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Are orthogonal polynomials always generated by a 2nd order recurrence relation?

Most of the orthogonal polynomials that we face, for example, in Physics are generated by recurrence relations of second order. Are there examples of orthogonal polynomials generated by higher order ...
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Simple multivariate orthogonal polynomials over the simplex

This is a soft question; I am looking for the most straightforward constructions of an orthogonal family of polynomials satisfying the following: multivariate (say in two variables), symmetric (if $p(...
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1answer
25 views

Efficient way to assemble polynomial terms with more than one variable?

Assume that I have a scalar variable $x$. For this variable, I can write down a second degree polynomial: $$f(x) = c_0 + c_1x + c_2x^2$$ where $c_0$, $c_1$, and $c_2$ are scalar coefficients. Now ...
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2answers
116 views

Orthogonal polynomials with respect to $e^{-|x|} \mathrm{d} x$ on the entire real line?

The Laguerre polynomials https://en.wikipedia.org/wiki/Laguerre_polynomials form a system of orthogonal polynomials with respect to the measure $e^{ -x} \mathrm{d} x$ on $(0,\infty)$. Is anything ...
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38 views

For which functions is $\sqrt{\int_1^2 f(x)^2*xdx} = 0$ true

I have to find all functions for which the following statement is true $$\sqrt{\int_1^2 f(x)^2*xdx} = 0$$ $f(x)$ can be anything: polynomial, trigonometric, exponential, ... I know I can leave the ...
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How to show orthogonality of the Laguerre polynomial $P_n(x)$?

At school, they ask me to solve this question: For $n \in \mathbb{N}$ and $x > 0$ we define $P_n(x) = \frac{1}{2\pi i}\int_{\Sigma}\frac{\Gamma(t-n)}{\Gamma(t+1)^2}x^t dt$ where $\Sigma$ is a ...
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Convergence of Gaussian-Quadrature measure

Suppose I have a measure $\alpha$ defined by $$ \alpha(x) = \sum_{i : \lambda_i < x} w_i $$ for nodes $\lambda_i$ and positive weights $w_i$, $i=1, \ldots, n$ (if we need $\alpha(x)$ to be ...
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The difference between f and the optimal function h* is orthogonal to the space the optimal function comes from - How can I interpret this visually?

I'm kind of confused about this theorem. I know and understand the proof, but I "can't see it". If I have a 2D function f, and it's optimal function is also a 2D function, thus they are from ...
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73 views

Approximating integral involving associated Laguerre polynomial

I need to numerically evaluate the following integral $$\sqrt{\frac{n!(n+1)!}{(n+\alpha)!(n+1+\alpha)!}}\int_0^\infty \frac{1}{\sqrt{x+c}}x^\alpha e^{-x}L_n^\alpha(x)L_{n+1}^\alpha(x)\;\mathrm dx$$ ...
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31 views

Problem with orthogonalizing the Laguerre polynomials

Alright, so I ran into a little problem while applying the Gram-Schmidt orthogonalization process. To the functions $\{1,x,x^2,x^3...\}$ over $x\in(0,\infty)$ with weight function $\sigma (x)=e^{-x}$. ...
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1answer
58 views

Laguerre Polynomial Termination

I had never learned much about Laguerre polynomials before, and I am trying to understand them for the first time. If we define the Laguerre equation as: $$xy'' + (1-x)y' + \lambda y = 0$$ Then if you ...
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1answer
59 views

Recursion relation for the Laguerre polynomials

How to come to the Laguerre recursion relation , $$(n+1)L_{n+1}^{(\alpha)}(x)+xL_n^{(\alpha)}(x)+ (n+\alpha) L_{n-1}^{(\alpha)}(x)=(2n+1+\alpha)L_n^{(\alpha)}(x) $$ from the sum for the generalized ...
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1answer
47 views

Roots of Hermite polynomials in closed form?

For some orthogonal polynomials, their roots can be expressed in closed form. For exemple, for the Chebychev polynomials of the second kind: $$ U_n(x) = \frac{\sin((n+1)\arccos(x))}{\sin(\arccos(x))} $...
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57 views

Function expansion in a basis of associated Legendre polynomimals

The Legendre polynomials $P_l(x)$ are complete in that any continuous function on $[-1,1]$ can be expanded as, $$ f(x) = \sum_{l=0}^{\infty} a_l P_l(x) $$ (see here for example). However, what is the ...
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What is the general solution of the Associated Legendre differential equation when A does not equal l(l+1)

I am sorry if this question isn't clear, I couldn't think of a better way to phrase it. I am a Physics student trying to solve the angular component of the wave function for a particle in a central ...
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1answer
49 views

Determining a measure based on recurrence relation for orthogonal polynomials

Suppose you were handed a sequence of polynomials $P_n(x)$ such that the $n$th one is degree $n$ and such that they satisfy a constant coefficient recurrence relation whose coefficients are taken from ...
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Given a sum of orthogonal polynomials $f(x)=\sum_{k=0}^{d}a_k P_k(x)=0$ where $x>0$, can I say that $a_k=0 \ \forall k$?

I think I can use the recursive argument that because the leading coefficient of the highest order polynomial $P_k$ is unequal to zero the corresponding $a_{d}=0$.
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Set of Orthogonal Polynomials over $[-a,a]$ That Has “Good” Approximating Power

I'm trying to approximate the function $f(x)=\max(-x,0)$ over $[-a,a]$ using polynomials. So, I used the Legendre polynomials for a generalized Fourier Series (relevant question), but with the ...
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35 views

Lagrange polynomial on $E_n$, projection on $E_{n-1}$

Let : $n \geq 1$ $E_n= \mathbb{R}^n[X]$ $\forall ~ 0 \leq j \leq n, \quad L_{n,k}(j)= \delta_{k,j}$ Scalar product $<P_1,P_2> =\sum_{k=0}^n P_1(k)P_2(k)$ Let $P \in E_{n-1}^{\perp}$ and $\...
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Orthogonal Polynomial series / orthogonal over finite extent / zero at origin / second derivative zero at limit of extent

I wish to observe trends in noisy data. Here are my assumptions regarding the physics of the data: Time series of values (finite in extent: $ S(t)$ discretized to $S_i$) Sampling is uniform in time ...
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1answer
56 views

Show that first order Chebyshev polynomials are in fact polynomials

Given the 1st order Chebyshev polynomials $$ G(x,t) = \sum_{n=0}^{+\infty} T_n(x) t^n = \frac{1-tx}{1-2xt+t^2} $$ I'm wondering how can I show that $T_n(x)$ are polynomials ?
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Can Laurent series be thought as an orthonormal series expansion?

Let $f:D\subseteq\mathbb{C}\to\mathbb{C}$ be an analytic function (except perhaps at the origin) defined over a disk $D$ centered at the origin. Its Laurent series on $D$ is $$ f(z) = \sum_{n=-\infty}^...
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Projection of a function onto the orthogonal complement of a subspace

I have a polynomial subspace, lets say $U$, which I applied the Gram-Schmidt algorithm to find an orthonormal basis. I had to find the projection of $cosh(x)$ onto $U$, so as I had found the ...
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11 views

Noise Effect on Autoregressive Polynomial Roots

So suppose I have the following signal: $$ x(t) = s(t) + n(t) $$ where $n(t) \sim \mathcal{N}(0, \sigma_{n}^{2})$ and $s(t)$ is an autoregressive process of order $P$. My question -- If I model $x(t)$ ...
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113 views

How to expand a harmonic function in terms of eigensolutions for bipolar / toroidal coordinates?

Consider the following harmonic function satisfying the Laplace equation in cylindrical coordinates, i.e., $\Delta f = 0$. $$ f(r,z) = \frac{z-h}{\left( r^2 + (z-h)^2 \right)^{\frac{3}{2}}} \, , $$ ...
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25 views

“Triple” Orthogonal Polynomials

Does there exist a generalization of the usual orthogonal polynomials where the usual $\int_a^b p_i(x)p_j(x)\text{d}x=\delta_{ij}$, is replaced by $\int_a^b p_i(x)p_j(x)p_k(x)\text{d}x=\delta_{ij}\...
3
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0answers
32 views

Properties for analogues of Bessel function

Consider the unit circle $\mathbb{S}^1 \subset \mathbb{R}^2$ and consider the uniform measure $\nu$ on $\mathbb{S}^1$ normalised so that $\nu (\mathbb{S}^1)=1$. The collection of functions $\{1,z,\...
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21 views

Zero's Counting Of Orthogonal Polynoms

First,we define the scalar product $ <P;Q> = \int_{I} P(t)Q(t)w(t)dt $ where $w(t)$ is a weight function. Then, we denote $ (P_n)_{n \in \mathbb{N}} $ the orthonormal basis of $(X^n)_{n \in \...
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50 views

How to solve this orthogonal matrix equation?

I have an equation like this: $ A = R^T * B * R $. Actually, the problem should be like this: $ argmin\sum_{i=0}^{n-1}(q_i^T*R^T*B*R*p_i)$. It's similar to the above equation. $q_i, p_i$ are known. $ ...

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