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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Show that $K(u)=\sum_{k=0}^r P_k(0) P_k(u) \mathbf{1}_{\{|u| \leq 1\}}$ is a kernel of order $r$
I have a question concerning the construction of kernels wit orthogonal polynomials. The instructor defined the orthogonal polynomials as
$$P_0(x)=\frac{1}{\sqrt{2}}, P_m(x)=\sqrt{\frac{2 m+1}{2}} \...
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A stereographic projection for the Chebyshev polynomials
This question may be too vague for the MSE crowd, if so, please feel free to ask clarifying questions or just remove.
The Chebyshev polynomials are a family of orthogonal polynomials typically defined ...
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Generating function of orthogonal polynomial basis
I'm studying the bases made up by orthogonal polynomial such as: Hermite, Legendre, Laguerre, Chebyshev. On my book there is a theoretical introduction that gives the difinition of generating function ...
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Given a polynomial, can we construct an orthogonal sequence?
Suppose we are given a polynomial $p(x)$ of degree $k$ which has all of its (distinct) roots in $[-1, 1]$. Is there is any way, generally, to choose a non-negative weight function $w(x)$ such that $\...
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Three-term recurrences in Orthogonal polynomials
hi I am reading one lecture note about Orthogonal polynomials (https://www.math.hkbu.edu.hk/ICM/LecturesAndSeminars/08OctMaterials/1/Slide2.pdf)
and there's one step in the proof in "Three-term ...
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Gegenbauer Polynomial Result
I am currently trying to familiarize myself with the Gegenbauer polynomials $\left(C_n^{\nu}(x)\right)$ and was reading [L Caffarelli, A. Friedman, Partial Regularity of the zero-set of solutions of ...
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How can one construct polynomial bases for positional estimation?
In the field of signal processing a popular problem is to try and estimate how something has moved as compared to a previous point in time.
In attacking this problem a long standing popular approach ...
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An improper integral involving a Gaussian and Hermite polynomials.
Let $ x \ge b \ge 0$ and let $\mu_0 \le 1/2$. Then let $j \in {\mathbb R}$. Consider the following integral:
\begin{equation}
{\mathcal J}^{(x,b,\mu_0)}_j := \frac{2^{\frac{j}{2}-1}}{\sqrt{\pi}} \cdot
...
- 9,860
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How to integrate products of Legendre functions over the interval [0,1]
The associated Legendre polynomials are known to be orthogonal in the sense that
$$
\int_{-1}^{1}P_{k}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{k,l}
$$
This is intricately linked to ...
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Why do cosines appear in the whitespace of plots of sines and/or cosines with regularly spaced frequencies?
If you plot many sines with regularly spaced frequencies over top one another, you will see cosines in the white space of the graph. Please see the image here. The graph is also particularly dark near ...
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Inner product of 4 Legendre Polynomials
Is there a closed form for the quadruple inner product of Legendre Polynomials such as:
\begin{align}
\int_{-1}^{1}P_k(x)P_l(x)P_m(x)P_n(x)dx
\end{align}
I am aware of solutions for the triple inner ...
- 23
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Looking for a function satisfying certain decay and smoothness properties
I am looking for a function $f : \mathbb{R} \to \mathbb{R}$ satisfying a few requirements.
$f(x) > 0$ for $x \in \mathbb{R}$.
$f(-x) = f(x)$ for $x \in \mathbb{R}$.
$f(x)$ is bounded as $x \to 0$.
...
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Laguerre polynomials and ill-defined Jacobi operators
I've been getting familiar with the theory of orthogonal polynomials, and one of the fundamental theorems that I'm working with states that a sequence of orthonormal polynomials $p_{n}(x)$ satisfies a ...
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Reference for generalization of the cosine function in 2 space for spectral method such that the basis is orthogonal?
So for a 1-d spatial, and 1-d time PDE,
I represent the solution as $U_N= \sum_{n=0}^N c_n(t) \phi_n(x)$,
where $\phi_n(x) = \frac{1}{\sqrt{\pi}}\cos(nx)$ and $x\in [-\pi,\pi]$
I was curious if anyone ...
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Orthogonal polynomials for a given weight function
I am seeking orthogonal polynomials which are orthogonal with respect to the weight $x^k$ for a given integer $k$,
$$
\int_a^b x^k \phi_m(x) \phi_n(x) dx = \delta_{nm}
$$
for some given $0 < a < ...
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approximating data using orthogonal polynomials
Recently, during numerical analysis lecture, I've learned about finding best approximation of element $f$ from normed linear space $E$, defined by $$\inf_{g\in \textbf{G}} \lVert f-g\rVert$$
where $G$ ...
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Is there an easy way to find orthogonal polynomials w.r.t. $w(x)=-\ln(x)$ on $(0, 1)$?
I've been evaluating them recurcively using $$\varphi_{n}=x^n - \sum_{k=0}^{n-1} \frac{\langle x^{k}, \varphi_{k} \rangle_{w}}{\langle \varphi_{k}, \varphi_{k} \rangle_{w}}\cdot\varphi_{k}$$
Where $\...
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Can we use zonal spherical harmonics to define the Gegenbauer polynomials?
For a fixed dimension $n$ and degree $k$, let $H$ be the space of all real homogeneous harmonic polynomials of degree $k$ in $n$ variables. We equip $H$ with the inner product $\langle f,g\rangle = \...
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"Easy" uniform bound for hermites functions.
The hermites functions are defined as $\psi_n(x) = (-1)^n(2^n n! \sqrt{n})^{-1/2} e^{x^2/2} \frac{d}{dx^n} e^{-x^2}$.
They satisfies many properties (see https://en.wikipedia.org/wiki/...
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Could someone help prove some properties of Legendre Polynomials?
I have already proved other properties of the Legendre polynomials, like:
$$P_n(-x) = (-1)^n \, P_n(x)$$
$$P_{2n+1}(0) = 0$$
$$P_n(\pm1)= (\pm1)^n$$
But I can't get this one:
$$P_{2n}(0) = \frac{(-1)^...
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How to prove Legendre Polynomials' recurrence relation without using explicit formula?
Assume there is an inner product on linear space $V = \{ \text{polynormials}\}$:
$$\langle f, g\rangle = \int_{-1}^1 w(t) f(t)g(t) dt$$ with $w(t) \ge 0$ and not identically zero.
Then we can ...
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Orthogonality of Chebyshev (Legendre) polynomials
I used the recurrence relation to construct Chebyshev polynomials
$$
T_0(x)=1
$$
$$
T_1(x)=x
$$
$$
T_{n+1}(x)=2x T_n(x) - T_{n-1}(x)
$$
for $x\in[-1,1]$
I found that
$$
T_0(x) \cdot T_1(x) =0$$
$$
T_0(...
- 1,481
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inner products on polynomials
So, I was studying about orthogonal polynomials and saw general examples of inner products on $\mathbb{R}[x]$, mostly of the forms
$$\langle f,g\rangle=\int f(x)g(x)q(x)dx,$$
for some kind of density $...
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Sequence of largest roots of Hermite polynomials
Let's denote with $\{x_n\}_{n\in N}$ the sequence of the largest root of the (statistical) Hermite polynomial $h_n$. Much is known about upper and lower bounds of the $x_n$, see for example here. ...
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Obtain a Sequence of Orthogonal Polynomials from their orthogonality relation.
I have been asked to show explicitly a Sequence of Orthogonal Polynomials that satisfy the following orthogonality relation
\begin{equation}
\displaystyle\int_{-\infty}^{\infty} H_m(x)H_n(x)e^{-x^...
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Reference(book or article) for an explicit formula of Legendre polynomials
The following explicit formula is stated for Legendre polynomials on Wikipedia.
\begin{equation}
P_n(x)=\sum_{k=0}^n {n\choose k}{n+k \choose k} \left(\dfrac{x-1}{2}\right)^2
\end{equation}
Do you ...
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Coefficients of orthogonal functions of the form sech * polynomial?
I took the SVD of part of a large set of data. Despite noise in the basis functions above ~20, the basis functions appeared to follow a pattern & looked interesting.
So, I found a sequence of ...
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Uniqueness of the nodes for Gauss-Legendre quadrature
Gauss-Legendre quadrature approximates $\int_{~1}^{1}f(x)dx$ by $\sum_{i=1}^nw_if(x_i)$.
Wikipedia says that
This choice of quadrature weights $w_i$ and quadrature nodes $x_i$ is the unique choice ...
user1095295
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orthonormal bases for the space of functions from $C^n \rightarrow C^m$?
For a single dimension, Legendre Polynomials and sets of sines and cosines find lots of applications. I've seen some of these extended to the space of functions $C^n \rightarrow C$.
Can these (e.g. ...
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Is the Taylor series a generalized Fourier series?
Let $V$ be the vector space of functions that are smooth $0$: they have derivatives of all orders. Every $f\in V$ can be (locally) expanded into a Taylor series, as $$f(x)=f(0)+f'(0)x+\frac{f''(0)}{2}...
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Integrals of products of Legendre polynomials
Define
$$
\tilde{P}_{n}^{m}(\tau)=(1-\tau^2)\frac{d}{d\tau}P_{n}^{m}(\tau)
$$
where $m$ and $n$ are integers such that $|m|\leq n$ and $P_{n}^{m}(\tau)$ are the associated Legendre polynomials. How do ...
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Recursion Relation for Coefficients of Orthogonal Polynomials
Given a measure on some manifold $d\mu(x)$, a set of orthogonal polynomials can be constructed through a Gram-Schmidt Orthogonalization Procedure
$$
P_n(x) = x^n - \sum_{k=0}^{n-1}\frac{\langle x^n, ...
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how to solve Poisson equation of inhomogeneous boundary condition by Galerkin method?
I'm working on solving 2D Poisson equation with source term f on irregular domain. Assuming I have a function u(x,y) which satisfy the inhomogeneous boundary condition, and I choose my Orthogonal ...
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Why orthogonal polynomial is orthogonal to all polynomials less degree?
Page 31 of https://www.math.ntnu.no/emner/TMA4215/2008h/lecture9.pdf defines a set of orthogonal polynomials
$$
\phi_k(x) = x \phi_{k-1}(x) - \sum_{j=0}^{k-1} \alpha_j \phi_j(x),
$$
starting from $\...
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Applications of Jack polynomials
I developed a Julia package for the computation of Jack polynomials. The zonal polynomials are particular cases ($\alpha=2$) of Jack polynomials (up to a renormalization), and they have some ...
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An example of a two-variate orthogonal polynomials
Let $m> n\geq 2$ be two coprime integers. Define
$$S:= \left\{\left(\frac{j}{m}, \frac{nj}{m}- \left\lfloor \frac{nj}{m} \right\rfloor\right), j =0,1,..., m-1\right\}\cup\{(1,1)\} \subset [0,1]\...
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Which polynomials/Fourier series satisfy this property?
I'm interested in finding polynomials $P$ for which there exists another polynomial $Q$ such that $\forall \theta$:
$$\| P(e^{i\theta}) \|^2 + \| Q(e^{i\theta})\|^2 = P(e^{i\theta})P^*(e^{i\theta}) + ...
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A continuous function on $[0,1]$ orthogonal to each monomial of the form $x^{n^2}$
Let us consider the continuous functions over $[0,1]$ fulfilling
$$ \int_{0}^{1} f(x) x^n\,dx = 0 $$
for $n=0$ and for every $n\in E\subseteq\mathbb{N}^+$. The Müntz–Szász theorem gives that
$$ \sum_{...
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Probabilistic Hermite Polynomials weight function and interval of integration
With the weight function $w(x)=\exp^{-\frac{x^2}{2}}$ and the interval of integration $[-\infty, \infty]$ one can obtain Probabilistic Hermite polynomials: $H_0(x)=1$, $H_1(x)=x$,$H_2(x)=x^2-1$, $H_3(...
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Expected value of dot product of orthogonal polynomials
I want to show that we can check if two orthogonal polynomials are orthogonal through computing the expected value of their dot product, but I'm not sure where to begin.
What I need is: $$ E[φ_{i}(X) ·...
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Recurrence relations for even orthogonal polynomials
I have been playing around with the theory of orthogonal polynomials, and it occurred to me that we might be able to build a family of orthogonal polynomials from even powers of a variable $x$. For ...
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Simple Identity for Derivative of Laguerre Polynomial
I'm working with Laguerre polynomials for numerically solving a differential equation, and I've stumbled upon an identity that I feel should be documented somewhere (e.g., https://en.wikipedia.org/...
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Gaussian Quadrature with Hermite Polynomials
I want to find a quadrature approximation of $\int f(x)e^{-\frac{x^2}{2}}$dx. Using the Hermite Polynomials 1, x, $x^2-1$, $x^3-3x$, $x^4 - 6x^2 + 3$. that is exact for polynomials up to order 7.
Now ...
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orthogonal polynomials and determinant of jacobi matrix
In the book 'Orthogonal Polynomials of Several Variables' by Charles F. Dunkl and Yuan Xu page 11 is the picture below. I assume the matrix in the picture is related to
Corollary 1.3.10 For the case ...
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Calculation of the integral of the Legendre polynomial of the second kind
Please tell me the possible options for calculating the integral of the form $\int\limits_{-\infty}^a\frac{Q_n(x)}{(x+b)^{n+2}}dx$, where $a\in[-2,-\infty)$; $b\in(-a,-\infty)$; $Q_n(x)$ - Legendre ...
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interlacing properties of zeros of orthogonal polynomials
Im working on the interlacing properties of zeros of orthogonal polynomials $p_n(x)$ - proved by Gabor Szegö (Orthogonal Polynomials - Theorem 3.3.2 )
The Theorem says:
Let $x_1 < … <x_n$ be the ...
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Are there radial basis functions $\phi_n(x)$ which can be factored as $f_n(x) x^{\alpha}$ for $0<\alpha<1$ and $n>0$?
I am looking for a set of radial basis functions $\phi_n(x)$ on $\mathbb{R}^+=[0;\infty[$ which satisfy some othogonality condition
$$ \int x \phi_n(x) \phi_{n'}^*(x) dx =\delta_{nn'}$$
and "...
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Show that for any monic polynomial of degree $n$, the inner product $\langle q,q \rangle \geq \langle p_n,p_n\rangle$
Let $\{ p_n \}$ be a family of monic orthogonal polynomials associated with a inner product $\langle f,g\rangle = \int_a^b w(x)f(x)\overline{g(x)}dx$. Show that for any monic polynomial of defree $n$, ...
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Prove that the Legendre polynomial recurrence relationship satisfies the defining differential equation
I am trying to show that from this recurrent relationship
$$ (n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x) $$
that the Legendre polynomial $P_n(x)$ satisfies the differential equation
$$ (1-x^2)P'' - ...
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Generating Hermite polynomial with coefficient recurrance relation algorithm
I am writing a math paper for my numerical analysis class about orthogonal Hermite polynomials. I want to implement the algorithm for generating the "probabilist's Hermite polynomials":
$$
\...
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