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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Hermite Polynomial and its Expectation

Currently, I'm stuck to some statement in a paper (in chapter 8: Nonlinear Model, from page 26 ~27). Although this topic generally covers statistics and machine learning theory, my main question is ...
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Lagrange interpolation and orthogonal polynomials

Suppose that $\{p_i(x)\}_{i=0}^{n}$ are pairwise orthogonal polynomials on the interval $[a,b]$, It means, $$ \int_{a}^{b} p_i(x)p_j(x)dx = 0\ , \;{i\neq j} $$ wherein $p_i(x)$ for all $i$ is a ...
schneiderlog's user avatar
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Hardy Hille type eigenfunction exapansion

I am trying a figure out Eigen-function expansion of the following kind. $$ \exp\left[{-\frac{1}{4\alpha} \left(x^2 + y^2\right)}\right] I_{l + \frac{1}{2}} \left(\frac{x y}{2 \alpha}\right) = \sum_{n}...
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Integral of product of Legendre polynomial and exponential function

Kindly help me with the following integral : $ I_l(a) = \int_{-1}^{+1} dx\, e^{a x} P_l(x) \quad $ ($a$ is real and positive). I thought to use the following relation given in Gradshteiyn and also ...
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Chebyshev polynomials orthogonal with respect to different weight function?

The following exercise appears in Ridgway Scott's Numerical Analysis: Where $\omega_n(x)$ is the Chebyshev Polynomial of the first kind, that is $$\omega_{n+1}(x)=2^{-n}\cos((n+1)\cos^{-1}(x)$$ I ...
modz's user avatar
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Closed form of $\int_{x}^{1}\left(1-s\right)^{-1/2}P_{m}^{\left(1/2,-1/2\right)}\left(s\right)ds$

Let $P_{m}^{\left(a,b\right)}\left(s\right)$ the $m$-th Jacobi polinomial. An old result of Bateman shows that $$ \left(1-x\right)^{\alpha+\mu}\frac{P_{m}^{\left(\alpha+\mu,\beta-\mu\right)}\left(x\...
Marco Cantarini's user avatar
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Orthogonal polynomials with convex weight [closed]

There is a well known fact, that if we have non-decreasing weight $w(x)$ in the finite interval $[a,b]$, then if $\{ p_n(x) \}$ is the set of corresponding orthogonal polynomials, the functions $|\...
lexxaedr's user avatar
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What does this specific notation mean?

In Bilinear generating functions for orthogonal polynomials (PDF) the following equation appears under Theorem 2.3.: How should I interpret the $j$-indexed denominator? I am familiar with the ...
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Orthogonal polynomial and determinant

I read a book about orthogonal polynomials written by Gabor Szego. There is a conclusion: Let the real-valued functions $f_0(x),f_1(x),\dots,f_l(x)$, where $l$ is finite or infinite, be of the class $...
Sxmx Simon's user avatar
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Two slightly different polynomial expansions for $\Xi(0)$. Could a connection between these two be derived?

After experimenting with polynomial expansions for the Riemann $\Xi(t)=\xi\left(\frac12+it\right)$-function, I landed on these two equations: with $M$ the KummerM confluent hypergeometric funcion and ...
Agno's user avatar
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Largest root of a linear combination of Chebyshev polynomials

I wonder if we can say something about roots of a linear combination of Chebyshev polynomials of the first kind. I have an example in my hand: $$(m+1)T_n(x)+(m-3)T_{n-2}(x)=0$$ for some $m>0$. I ...
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Construction of orthogonal matrices from orthogonal polynomials

Let $\{P_n\}_{0\le n\le N}$ be the family of orthogonal polynomials associated with the following inner product: $$ \langle f, g \rangle_{1} = \sum_{k=0}^{N}{f(k)g(k)w_1(k)} $$ where $ w_1(\cdot)$ is ...
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A uniform (non-asymptotic) upper bound for Hermite polynomials in the complex plane

In a paper dating back to 1990, Eijndhoven and Meyers [1] mention the following "elementary" upper bound for Hermite polynomials on the whole complex plane: $$ \forall z \in \mathbb{C}, \...
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Existence of functions with orthogonality property w.r.t polynomial

Does there exist a set of functions $\{f_j\}$ and real numbers $a,b$ (potentially dependent on $i,j$) which have the property $\int_{a}^{b} x^i f_j(x) dx =A \delta_{ij}$ for some constant $A$? I don't ...
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Density of Chebyshev nodes

While reading some notes, I came across the following statement: ``Chebyshev points have density $\mu(x) = \frac{N}{\pi\sqrt{1-x^2}}$". I would like to understand where this formula comes from. ...
Okazaki's user avatar
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Getting Modified Zernike Polynomial (Radial Part) ODE from Jacobi Polynomial ODE

Question: The Jacobi differential equation in terms of the Jacobi polynomial $P_{n}^{(\alpha,0)}(x)$ is given by: \begin{equation} (1 - x^2) P_{n}^{(\alpha,0)''}(x) + (-\alpha - (\alpha + 2) x) P_{n}^...
Hamit's user avatar
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best polynomial approximations to $f$ vanish at $0$ imply $f$ is an odd function

It is known that if $f\in C[-1, 1]$ is odd/even, then the best polynomial approximation (in $L^{\infty}$ norm) of degree $n$, denoted by $p_n$, must also be odd/even. This has been asked on MSE before....
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hyperconfluent geometric function expressed in Laguerre polynomials references

I am searching for references that express the relation between hyperconfluent geometric function of the first kind ${}_1F_1\left(a, b; z\right)$ expressed in Laguerre polynomials for example a ...
Math Explorer's user avatar
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Product of d-dimensional Legendre polynomials

Let $P_n:\mathbb{R}\rightarrow\mathbb{R}$ be the $d$-dimensional Legendre polynomials, that is they are orthonormal polynomials w.r.t the probability measure $\mu_d$ on $[-1,1]$ given by $\mu_d= \...
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The $n$-th reproducing kernel

Let $N$ be a non negative integer. The sequence of monic orthogonal polynomials $\{P_n(x)\}_{n}$ respect to the inner product $$ \langle f , g\rangle =\sum^{N}_{k=0}{f(k)g(k)\rho(k)} $$ With $\rho$ ...
Karim's user avatar
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Understanding the Dual Emergence of Legendre Polynomials in Differential Equations and Orthogonalization

I am currently examining the mathematical properties of Legendre polynomials and have observed their emergence in two distinct areas: as solutions to a specific class of differential equations (...
Hakan Akgün's user avatar
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Gram-schmidt process for polynomials [closed]

$$S = \{1, x , x^2\}$$ S is a set of orthogonal vectors. So instead of applying the gram-schmidt process to obtain the orthonormal basis, can't we just do $$S' = \{1/||1||, x/||x||, x^2/||x^2||\}$$ ...
Samyak Jain's user avatar
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Divergence of Mehler's Hermite polynomial series

According to Mehler's formula, $$ \sum\limits_{n=0}^{\infty}\left(\cfrac{\rho}{2}\right)^n\cfrac{H_n(x)H_n(y)}{n!}=\cfrac{1}{\sqrt{1-\rho^2}}\exp\left(-\cfrac{\rho^2(x^2+y^2)-2\rho x y}{1-\rho^2}\...
george_ch's user avatar
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Simplify this sum of Bell polynomials similar to the generating function

During my research, I encountered this expression: $$ \sum_{p=0}^{\infty} \frac{1}{(p+1)!} \sum_{k = 0}^{p} k! \, u^k \, B_{p,k}(f_1, \dots, f_{p-k+1}) $$ with the arguments of $B_{p,k}$ being the ...
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Charlier-Sobolev-type orthogonal polynomials

I am currently reading an article, and I am a bit stuck on understanding a sentence in this article. The sentence, as indicated in the image, is: " In the case of Charlier polynomials, the ...
Made's user avatar
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Bounds on complex Hermite polynomials

There is a well-known bound on Hermite polynomials: $$ \left|\cfrac{H_n(x)}{\sqrt{2^n n!}}\right|\le e^{0.5x^2}, $$ where $x$ is real. I am trying to find a bound of the following form: $$ \left|\...
george_ch's user avatar
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how to prove the following or a reference for the following

Let $L_n^k(x)$ be the Laguerre polynomial of type $k$. I need a reference of this forumla $\sum_{n=0}^{\infty} \frac{n !}{(n+a) \Gamma(n+k+1)} L_n^k(x) L_n^k(y)=\frac{1}{\Gamma(k+1)} \Phi(a, k+1 ; x)...
Ryo Ken's user avatar
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Addition formula for generalized Laguerre polynomials

For the Hermite polynomials, there is the following addition formula Is there a similar formula for the generalized Laguerre Polynomials, in particular for $a=b=1/2$. I.e. what is $L^m_k(0.5x + 0.5 y)...
user2224350's user avatar
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Gegenbauer functional equations

Let $C_n^\lambda$ be the Gegenbauer Polynomial with the parameter $\lambda$. We have the defining recurrence relation $$nC_n^\lambda(x)=2(n+\lambda-1)x\,C_{n-1}^\lambda(x)-(n+2\lambda-2)C_{n-2}^\...
Sellerie's user avatar
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Biorthogonal sequence for monomials in L2(0,1)

A sequence $(x_n^*)_{n \in \mathbb{N}}$ is a biorthogonal sequence for a sequence $(x_n)_{n\in\mathbb{N}}$ in a Hilbertspace $H$ if $$ (x_m^*, x_n) = \delta_{mn} $$ for all $m,n \in \mathbb{N}$. Does ...
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Question about multidimensional polynomials basis develoment

Working on a physics branch that make use of (multidimensional) Hermite polynomials, I struggle to find an accurate insight about expansions on this basis. Let me explain: Consider a function $f : \...
Atmos's user avatar
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2 votes
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Alternative expressions for Krawtchouk (Kravchuk) polynomials

For fixed non-negative integers $n$ and $q \geq 2$, the $k$-th Krawtchouk (Kravchuk) polynomial is defined as $$K_k = \sum_{j=0}^k (-1)^j (q-1)^{k-j} \binom{X}{j} \binom{n-X}{k-j} \in \mathbb{Q}[X]$$ ...
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When are spherical harmonic expansions valid?

It is known that a square integrable function on the sphere can be expanded in a basis of spherical harmonics, $$ f(\theta,\phi) = \sum_{l=0}^{\infty} \sum_{m=-l}^l c_l^m Y_l^m(\theta,\phi) $$ where $\...
vibe's user avatar
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2 votes
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Kernel Polynomials and Extremality

I am trying to prove the following theorem: Theorem Let $x_0 \in \mathbb{R}$ and $q_n(x)$ a polynomial of degree at most n, normalized by the following condition: $\int_a^b (q_n(x))^2 w (x)dx$=1. The ...
babu's user avatar
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5 votes
1 answer
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Proving that quotient of orthogonal polynomials is a Padé approximant of Stieltjes transform.

Conext and notation. In the question below, $R_n(z) / S_n(z)$ denotes the n-th convergent of the continued fraction $$ \frac{1|}{|z-b_1} + \frac{-a_2|}{|z-b_2} + \dots + \frac{-a_n|}{|z-b_n} + \dots,$$...
xyz's user avatar
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Proving the reproducing property of kernel polynomials

I need to prove the following property related to kernel polynomials: $\int_a^b K_n(t,x)q_n(x)w(x)dx=q_n(t)$, where $q_n(x)$ is a polynomial of degree less or equal to $n$, $w(x)$ is a weight function ...
babu's user avatar
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Finding the minimum value of an integral using least squares-mean function approximation

I am starting to study some concepts related to orthogonal polynomials and my teacher told me to prove the following theorem, Theorem The integral $\int_{a}^{b} Q_n^2(x)w(x) dx$ where $Q_n(x)$ is any ...
babu's user avatar
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A bound for error of least square

I'm approximating $f(t)$ by $P_n(t)$, a real polynomial of arbitrary degree $n$, by minimizing the $L_2$ norm of its difference in $[0,1]$ $$R_n=|| f(t)-P_n(t)||_2^2=\int_0^1(f(t)-P_n(t))^2\,dt$$ ...
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Implication of dot product $(Q_n(x), 1)\equiv \int_a^b {p(x)Q_n(x)dx} = 0$ in $L_2[a,b,p]$, where $Q_n(x)$ is a polynomial of $n$-th degree.

Dot product $(Q_n(x), 1)\equiv \int_a^b {p(x)Q_n(x)dx} = 0$ in $L_2[a,b,p]$, where $Q_n(x)$ is a polynomial of $n$-th degree. According to my textbook this happens only if $Q_n(x)$ has at least one ...
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Jacobi Orthogonal Polynomials - Scipy (Python) vs My Recurrence Implementation

I implemented the Jacobi orthogonal polynomial recurrence relation using the following: Source: http://lsec.cc.ac.cn/~hyu/teaching/shonm2013/STWchap3.2p.pdf However, when I compare to the Scipy (...
Colton Campbell's user avatar
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2 answers
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Proving that the composition of orthogonal transformations is orthogonal in a finite dimensional inner product space $V$.

Given $S$,$T$ orthogonal transformations in an inner product space $V$ over $\mathbb R$ such that $ \dim(V) < \infty$. Also given that $v \in V$ | ($S$$\circ$$T) (v) = S(T(v))$ I need to prove that ...
dy 1995's user avatar
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How do I find the leading coefficient of a Legendre polynomial?

I'm trying to construct the Legendre polynomials from the differential equation. As is done in this set of lecture notes, I can get an expression for the coefficient $c_{l-2k}$ in terms of $c_l$: $$c_{...
anna_nimmus's user avatar
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Calculating the orthogonal projection for polynomial functions

The task is: Let ${E := {ax^3+bx^2+cx+d : a, b, c, d ∈ R}}$ be the vector space of all real polynomials of degree at most 3 and let ${F}$ be the subvector space of all real polynomials of degree at ...
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why no initial conditions are required in the differential equation/eigenfunctions problem of orthogonal polynomials?

in section 4.2 of the book "Special functions, a graduate text" says the following: We return to the three cases corresponding to the classical polynomials, with interval $I$, weight $w$, ...
CACM6's user avatar
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Some generalised combinatorial formula concerning the integral over a square of Gegenbauer Polynomials

Two weeks ago I asked this question on here to figure out a way to prove Rodrigues' formula for Gegenbauer Polynomials without going via Jacobi Polynomials or the orthogonality criterion. I didn't ...
Sellerie's user avatar
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Stieltjes transformation of an even measure

I am confused about the behavior of the Stieltjes transform $s_\mu(x)$ for an even measure $d\mu(x)$. Let $d\mu(x)$ be positive and even on $[-1,1]$; the Stieltjes transform is the function $$ s_\mu(z)...
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Rodrigues's formula via Lagrange Inversion

I am currently working with Gegenbauer/ultraspherical polynomials, which can be defined by their relation to Jacobi polynomials: $$ C_n^\lambda(x)=\frac{(2\lambda)_n}{\left(\lambda+\frac{1}{2}\right)...
Sellerie's user avatar
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0 answers
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Does distributional identity for Hermite polynomials imply anything pointwise?

The distributional identity says $\sum_{k=0}^\infty \frac{1}{\sqrt{\pi} 2^k k!}e^{-x^2/2-y^2/2}H_k(x)H_k(y)=\delta(x-y)$. For my understanding, that means for any nice function $g$, \begin{equation*} \...
Frht's user avatar
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Vanishing integral on $(-c,c)$

Assume we have a function $f\colon (-c,c)\to\mathbb{R}$ such that $\int_{-c}^cf(s)\,\mathrm{d} s=0$. Is there a general approach to finding conditions on $f$ that would ensure that $\int_{-c}^c s\cdot ...
Sellerie's user avatar
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1 answer
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$x \cdot F_n$ is a linear combination of $\{ F_{n−1}, F_n, F_{n+1} \}$ for every positive integer $n$.

Let $F_n$ denote the $n$th polynomial obtained through Gram-Schmidt orthogonalization applied to the sequence $$\{ 1, x, x^2, x^3 , \cdots , x^n \}$$ with the inner product $$ \int_0^1 f(x) \cdot g(x) ...
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