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

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Linear Algebra - Weighted Inner Product of Polynomials [on hold]

Given the weighted inner product $\langle f,g\rangle = \int^1_{-1}f(x)g(x)x^2dx$ How do you find an orthogonal basis of the space $\Bbb P^1$ of polynomials of degree $\le$ 1. And how do you find the ...
0
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1answer
18 views

Polynomial Chaos: How are the pdfs calculated from the response surface?

Lets assume one has the following response surface: $y(x,\xi) = \sum^N_{i=0} c_i H_i(\xi)$. Where $\xi$ is Gaussian and $H_i$ is the $i^{th}$ Hermite polynomial. I've seen a lot of papers show the PDF ...
1
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1answer
19 views

Applying Gram-Schmidt process to a set of vectors to find first three polynomials orthogonal with respect to inner product

$.$ $\langle f, g \rangle = \displaystyle\int_{-1}^{1} f(x)g(x)dx$ Apply Gram-Shmidt process to the set of vectors $:$ {1, x, $x^2$, ...} to find the first three polynomials orthogonal with respect ...
0
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2answers
28 views

Mutually orthogonal set of vectors

Show that the standard basis: $$..$$ $\mathscr{B} = \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0\\ 1 \\ 0 ...
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0answers
14 views

Give the idempotent generators of the four binary QR codes C1 , C2 , C3 , C4 , of length 7.

I'm having trouble on some homework. This is the last problem and I can't figure it out. Can anyone help or point me in the right direction? Thanks! For each code Ci , 1 ≤ i ≤ 4, from part (a), give ...
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0answers
13 views

Normalization constant hn of Hn(x)

Deduce the normalization constant $h_{n}$ of the hermite Polynomial $H_{n}(x)$ from below function. $\sum_{n=0}^{\infty}\frac{(st)^n}{(n!)^2}\int_{-\infty}^{\infty}e^{-x^2}H_{n}^2(x)dx$ Work done ...
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13 views

extension of trigonometric functions as basis functions to higher dimensions

Trigonometric functions forms an orthonormal basis functions for $L^2[a,b]$, with corresponding normalization coefficients. I want to know if this result can be extended to higher dimensions. For ...
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17 views

Complex orthogonal polynomial

If we take a complex inner product over $\mathbb{C}[Z]$ we can construct a family of orthogonal polynomial $(P_n(Z))_n$, my question is "for each n the zero of $P_n(Z)$ are simple, like the reel case ...
0
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1answer
18 views

Series identity of Laguerre polynomials

This came up in the computation of an ensemble average in quantum mechanics. According to Mathematica, we have the curious identity \begin{equation} \sum_{n=0}^\infty \exp(-bn)L_n(2a) = ...
0
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1answer
24 views

Orthogonal polynomials derive normalization factor

I have the following problem, and have literally no idea where to start! Even a hint to get me going would be apreciated. I believe the question is concerning Laguerre polynomials. Let $\alpha$ > ...
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0answers
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Integral of product of Hermite functions with rescaled weights.

Let $$h_{k}(x)=c_{k}(-1)^k e^{\frac{x^2}{r^2}}\frac{d^k}{dx^k}e^{-\frac{x^2}{r^2}}$$ be the standard Hermite polynomials, rescaled with a given parameter $r>0$. The normalizing constant ...
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0answers
12 views

If I use a Hermite-Gauss function as a basis which functions can I represent?

I know that Hermite polynomials are orthogonal with eachother as follows: $$\langle H_n,H_m \rangle=\int_{-\infty}^\infty H_n(x) H_m(x) \exp(-x^2) \,\mathrm dx$$ If I define a basis function (the ...
1
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1answer
34 views

How to compute the inner product of a Hermite polynomial with itself

I am trying to prove the well-documented fact that if $H_{n}(x)$ is the $n$th Physicists Hermite polynomial then: $$\left\langle H_{n} \middle| H_{m} ...
1
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1answer
33 views

Using my orthonormal basis to find a polynomial that best approximates the polynomial $t^3$

I want to find the second-order polynomial that best approximates $t^3$, with respect to the norm of the vector space $V$. I first proved the bracketing map given in the problem was indeed an inner ...
0
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0answers
24 views

Use orthogonal polynomials to find&fit probability distribution to missing data.

Following the excellent discussion in a previous question, I need to find out if I can know which is the "best" PDF from a pool of candidate PDFs to fit a series with missing data elements. The ...
2
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2answers
77 views

An infinite series involving Legendre polynomials

For $x \in [-1,1]$ and $0 \le g < 1$, consider the convergent series $$ H(x,g) = \sum_{k = 0}^\infty (2k+1) g^k P_k(x)^2 $$ where $P_k$ is the $k$-th Legendre polynomial. Then $H(1,g) = ...
0
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1answer
26 views

Orthonormal Basis for $[5,1+t]^{\perp}$

Consider the vector space $\Bbb{V}=P_3(\Bbb{R})$ of the real polynomials of degree less or equal 3, with the inner product given by $$\langle f,g\rangle=\int_0^1f(t)g(t)dt,\forall f,g\in\Bbb{V}.$$ ...
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2answers
34 views

Matching polynomial of a complete bipartite graph is a generalized Laguerre polynomial.

Consider a graph $G$ with $n$ vertices. Let $m_k$ be the number of $k$-edge matchings. (A matching in a graph is a set of edges without common vertices.) Several different types of matching ...
3
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1answer
43 views

Show that Polynomials Are Complete on the Real Line

Consider the Hilbert Space of weighted-square-integrable functions f(x): $$ \begin{equation} \int_{-\infty}^{\infty}\frac{f(x)^2}{e^{x}+e^{-x}}dx<\infty. \end{equation} $$ Note this integral is ...
0
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1answer
43 views

Can anybody explain orthogonal basis for constant polynomials?

I have a problem. $(f,g)= \int_{1}^{3} f(x)g(x) \, dx$ and $f(x)=\dfrac{1}{x}$ I have to find out a nearest constant polynomials to $f$. So, I assumed $a$ to be a orthonormal basis for constant ...
0
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0answers
33 views

LCM of two polynmials when they are represented as point-value.

I`m wondering if we can obtain least common multiple of two polynomial when each polynomial represented as point-value. To be more clear, can we do any computation on these point-values and obtain ...
0
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1answer
48 views

Why are non-polynomial Legendre functions and Legendre polynomials not orthogonal?

The eigenfunctions of distinct eigenvalues for a Hermitian operator are proved to be orthogonal. Why does the same not apply to Legendre polynomials and functions that have different eigenvalues ? ...
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32 views

Iterated integrals of Legendre polynomials

Let $P_n$ denote the $n$th order Legendre polynomial. It is known that for $n\not= 0$ we have $\int P_n(x)\, \mathrm{d}x = \frac{1}{1+2n} (P_{n+1}(x) - P_{n-1}(x))+ C $. Setting all integration ...
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0answers
21 views

How can I expand a zonal polynomial in a sum of two matrices? C(A+B) = C(A)*C(B)

I am trying to solve an integral involving zonal polynomials. TO do that I need to somehow separate out the zonal polynomial C(A+B) where my A varies but B is constant. They're all square matrices ...
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0answers
34 views

Example of incomplete orthogonal system in L2 space

I was asked to provide an example of an incomplete orthogonal in any L2 space and was wondering if this is a valid answer. Take some known complete system, for example the Hermite polynomials over ...
3
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0answers
43 views

Gaps between roots of trigonometric polynomials?

Given a polynomial in $e^{\mathrm{i}k t}$ of the form $$ p(t) = \sum_{-n\leq k\leq n} c_k e^{\mathrm{i}k t} $$ with $\bar c_{-k} = c_k$, is there a good way of characterising how close its roots can ...
5
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1answer
57 views

Asymptotics bound on Jacobi polynomials in the complex plane and for large $n$

Dear mathematicians and theoretical physicists, I am a theoretical physicist and I am bothering to you since I need to know some asymptotic and analytical properties of Jacobi polynomials ...
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2answers
53 views

Least squares approximation: Legendre polynomial

Find the best quadratic least squares approximation to $f(x)=e^x$ on $[-1,1]$ with respect to the inner product $\langle f(x),g(x) \rangle = \displaystyle\int_{-1}^1 f(x)g(x)dx$. I cannot figure out ...
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1answer
42 views

Proof that Qm(x) and Pn(x) have the same sign

I have difficulty in this exercise, is from the book Apostol Calculus Vol 2, I really want to know is there is a technique that can show that the two polynomials share the same roots. Thanks.
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95 views

Proving that two functions involving integrals with Legendre polynomials are equal

I have two functions that I expect to be equal (where $P_{2l}$ are the even Legendre Polynomials): $$F_{2l}(x)=x\, \tanh(\pi x/2)\left|\int_0^1 u^{i x-1}P_{2l}(u)\,du\right|^2$$ ...
5
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2answers
270 views

Curious gamma identity

I found the following curious identity for the gamma function on Wikipedia for which I'd like to know some references (proof, history, etc). The identity is as follows: $$\Gamma(t) = x^t ...
1
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1answer
34 views

A sum of Laguerre polynomials

I'm looking to find a closed-form expression for the sum $$S = \sum_{n=0}^N e^{-x/2} L_n^{0}(x),$$ where $L_n^{0}$ is the $n$th Laguerre polynomial. Using the formula $$L_n^{\alpha}(x) = \sum_{m=0}^n ...
1
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1answer
234 views

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 ...
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0answers
14 views

Best uniforme approximation of nule function in the meaning of Tchebychev

I would be interest to know , why exactly approximate a nule function and it is in the same time nule ? I would be like someone give me enough (papers, link ...) about "The best uniforme ...
1
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2answers
33 views

not complete polynomial and roots

It might be very simple but I need a formal proof for accepting or rejecting the idea below. Let g be a polynomial of the order of n given below $$ g(L)=1-\theta_1 L-\theta_2 L^2- \ldots -\theta_n ...
11
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1answer
182 views

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 ...
0
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2answers
398 views

Legendre polynomial expansion of the unit step function.

The problem is to determine the expansion of the unit step function in terms of Legendre polynomials on the interval $[-1,1]$. Here the Legendre polynomials are the family of orthogonal polynomials ...
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0answers
27 views

Orthogonal polynomials induction proof

I tried writing this all out but cannot seem to get anything sensible. Basically I want to prove that assuming w(x) is the weight function of a Gram Schmidt orthogonalization process and w is an ...
0
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0answers
83 views

Laplace transform and Laguerre Polynomials

let be the Laplace transform of a function $$ F(s)= \int_{0}^{\infty}dtf(t)e^{-st} $$ then if the function satisfies that for every integer 'n' $ D^{n}f(x)=0 $ then $$ s^{n}F(s)= ...
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0answers
19 views

orthogonal polynomials and quadrature formulae

is it true that for every set of orthogonal polynomials $$ \int_{a}^{b} w(x)P_{n}(x)P_{m}(y) =a_{m,n}\delta _{m}^{n}$$ with respect a measure $ w(x)$ can we always find a quadrature formula for ...
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25 views

Finding another solution in certain form (confluent hypergeometric series)

Consider the differential equation $$xf''(x)+(\alpha+1-x)f'(x)+nf(x)=0 ..........(*)$$, we know that the Laguerre Polynomial ...
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0answers
24 views

Checking a solution of differential equation

I am trying to prove that the Laguerre polynomial $$L=L_n^{\alpha}(x)=\frac{x^{-\alpha}e^{-x}}{n!}\frac{d^n}{dx^n}(e^{-x}x^{n+\alpha})= ...
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1answer
71 views

Fermat Last theorem on Poly-Euler numbers

The poly-Euler numbers, denoted as $E_{n}^{(k)}$, are defined by the following generating functions :$${2\operatorname{Li}_k(1-e^{-x}) \over 1+e^{-x}}=\sum_{n=0}^\infty E_n^{(k)}{x^n\over n!}$$ The ...
0
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1answer
61 views

Use the zeroes of T3 to construct an interpolating polynomial

Use the zeroes of T3 to construct an interpolating polynomial of degree two for the function x^3 on the interval [-1,1] Okay, so I have been looking at Finding the zeroes using Chebyshev polynomials ...
4
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1answer
99 views

Numerical evaluation of polynomials in Chebyshev basis

I have high order (15 and higher) polynomials defined in Chebyshev basis and need to evaluate them (for plotting) on some intervals inside the canonical interval $[1,\,-1]$. A good accuracy near 1 and ...
2
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1answer
58 views

Orthogonality of Lagrange Polynomials in Hermite Inner Product

My question is as shown above. I have churned through the first part, but I am stuck on showing the orthogonality of the Lagrange polynomials. My first hope was to use the fact that the Lagrange ...
4
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1answer
253 views

functions orthogonal to the exponential Bell polynomials

Consider the single variable Bell polynomials $\phi_{n}(x)$ given by: $$\phi_{n}(x)=e^{-x}\sum_{k=0}^{\infty}\frac{k^{n}x^{k}}{k!}$$ I am looking for a set of functions $\tilde{\phi}_{n}(x)$ such ...
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45 views

Turning a summation into an integral

I have a summation of the form: $$y(x) = \sum\limits_{h=-L}^L\frac{A(h)\cdot R(h)^2}{((x-h)^2+R(h)^2)^{3/2}}$$ Where I wish to solve/optimise $R(h)$ (leaving $A(h) = const/h$) or $R(h)$ and $A(h)$ ...
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1answer
42 views

How could I prove that $P_n (1)=1=-1$ for the Legendre polynomials?

I could "prove" that $P_n(1)=1=-1$. But I'm not sure where is my mistake. If I use the generating function: $$\frac{1}{\sqrt{1-2z+z^2}}=\sum_{n=0}^{\infty}P_n(1)z^n$$ Since: $\sqrt{1-2z+z^2 } = ...
0
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0answers
27 views

a question about Jacobi polynomials

Imagine if I have a defined function $\omega(\alpha, \beta, \gamma)$, where $0<\alpha<2\pi$, $0<\beta<\pi$, and $0<\gamma<2\pi$. I can then expand this function into series just like ...