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

# Questions tagged [orthogonal-polynomials]

660 questions

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### Roots of Legendre Polynomial

I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials.
Are the roots always simple (i.e., multiplicity $1$)?
Except for low-degree cases, the roots can't be…

user3180

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**24**

votes

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### Local maxima of Legendre polynomials

When I plotted the (normalized) Legendre polynoials, I couldn't help noticing that all the local maxima lay on a really nice curve:
What is the equation of the curve (and how can we arrive to that equation)?

Dave

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### Scalar product and uniform convergence of polynomials

Given two functions $u$ and $v$, you can compute $(u|v) = \int_0^1 (u(t)'-u(t))(v(t)'-v(t)) \,\mathrm{d}t$. It resembles the scalar product $\int_{-1}^1 u(t)v(t) \,\mathrm{d}t$, which leads to Legendre polynomials, with a major difference: while the…

Jean-Claude Arbaut

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### The roots of Hermite polynomials are all real?

The Hermite polynomials are defined as $$H_n(x)=(-1)^n e^{x^2}\dfrac{d^n}{dx^n}e^{-x^2}.$$
How does one prove that all the roots of the Hermite polynomial are real?

89085731

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### Hermite polynomials recurrence relation

Hermite polynomials $H_n (x)$ can be obtained using the recurrence relation
$$H_{n+1} (x)=2xH_n (x)-2nH_{n-1} (x).$$
To prove this, I started by calculating the first derivative of the Hermite's Rodrigues formula $H_n (x)=(-1)^n e^{x^2} …

Glenio Rosario

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### What does orthogonality mean in function space?

The functions $x$ and $x^2 - {1\over2}$ are orthogonal with respect to their inner product on the interval [0, 1]. However, when you graph the two functions, they do not look orthogonal at all. So what does it truly mean for two functions to be…

Chinedum Ukejianya

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### Connection between Hermite & Legendre polynomials

Prove that $$H_n(x)= 2^{n+1}e^{x^2}\int_x^\infty e^{-t^2}t^{n+1}P_n\left(\frac{x}t\right)dt,$$
where $H_n$ is Hermite polynomial & $P_n$ is Legendre polynomial

rasha kam

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### Isomorphisms of inner-product spaces

I think I understand why all finite-dimensional vector spaces over a field $\mathbb{K}$ are isomorphic to $\mathbb{K}^n$. Any linear map $T: V \rightarrow W$ between finite-dimensional vector spaces taking a basis to a basis is automatically an…

AndrewG

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### On the completeness of the generalized Laguerre polynomials

I am trying to prove that the generalized Laguerre polynomials form a basis in the Hilbert space $L^2(\mathbb{R})$.
1. Orthonormality
\begin{equation}
\int_0^{\infty} e^{-x}x^kL_n^k(x)L_{m}^k(x)dx=\dfrac{(n+k)!}{n!}\delta_{mn}
\end{equation}
2.…

QuantumMechanic

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### Intuition of orthogonal polynomials?

Recently, while learning about the regularization methodologies(in machine learning), i came across orthogonal polynomials ( of Legendre's polynomials), I looked up on the Internet and there are formulas for the same, but i didn't get any intuition…

supermeat boy

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**12**

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### Legendre Polynomials Triple Product

I have to solve the following integral:
\begin{align}
\int_{-1}^{1} \left(x^2 -1\right)^3 P_k(x)\,P_l(x)\, P_m(x) \;dx
\end{align}
where $P_{k,l,m}$ are Legendre Polynomials
The triple product
\begin{align}
\int_{-1}^{1} P_k(x)\,P_l(x)\, P_m(x) \;dx…

Matt

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### Is there a representation of an inner product where monomials are orthogonal?

There are plenty of examples of inner products on special sequences of polynomials such that they are orthogonal. I can't quite wrap my head around the inner product s.t. monomials are orthogonal. Say we have polynomials defined on the unit…

muaddib

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### Will this sequence of polynomials converge to a Hermite polynomial pointwise?

While trying to solve this question my testing lead to an observation that I found interesting in its own right. Consider the linear transformation $L:P\to P$ from the space of polynomial functions $p\in\Bbb{R}[x]$ to itself defined by setting…

Jyrki Lahtonen

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### How does one prove Rodrigues' formula for Legendre Polynomials?

I am trying to prove that $\frac{1}{n!\space2^n}\frac{d^n}{dx^n}\{(x^2-1)^n\}=P_n(x)$, where $P_n(x)$ is the Legendre Polynomial of order n.
I've been told that the proof uses complex analysis, of which I know nothing, isn't there a proof with…

Alubeixu

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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 precisely, when $N$ is even, the sum is positive…

Lucky

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