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

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Chebyshev representation of polynomial

In Carl de Boor's A Practical Guide to Splines (1978) problem II.3.a demands a proof that a polynomial $P_ng$ of order $n$ which agrees with a function $g:\mathbb{R}\rightarrow\mathbb{R}$ at $\tau_1, ...
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2answers
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An identity involving the Chebyshev polynomials

Let $n \in \{0, 1, 2, \dots\}$ and let $T_n$ denote the Chebyshev polynomial of degree $n$: $T_n(x) = \cos\left(n \arccos(x)\right)$. Let $t_0, t_1, \dots, t_n$ be $T_{n + 1}$'s roots: $t_i = ...
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Integral of Legendre polynomials

Is there any way of analytically simplifying the integral \begin{equation} \int_{-1}^1 (1-x^2)^{n+k+7/2} P_{2n+1}^1(x) P_{2k+1}^1(x) \, dx, \end{equation} where $P_l^m(x)$ is the associated Legendre ...
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1answer
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Gram-Schmidt procedure on functions

I have been applying the Gram-Schmidt procedure with great success however i am having difficulty in the next step, applying it to polynomials. Here i what i understand If i have 2 functions, say ...
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Spectral convergence for collocation methods

Spectral methods work (simplified) as follows. Consider the problem \begin{align} \partial_t u(t,x) = \mathcal{L} u(t,x) \end{align} where $\mathcal{L}$ is some differential operator. We then try to ...
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1answer
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Finding a vector for an orthonormal basis after using gram-schmidt.

So I started with a basis for $P_3$ (polynomial of degree less than 3). $$\{1,x,x^2\}$$ which has inner product define as $$\langle p,q \rangle=p(-1)q(-1)+p(0)q(0)+p(1)q(1)$$ For this product I found ...
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Find a realizable set of moments from a space defined by Hankel determinants

I have a question regarding the Hankel determinants. Let us consider a case where we have a set of moments. We want to see if these moments sets provides the positivity of the Hankel determinants ( ...
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q-Hermite polynomials

It is well known that the q-Hermite polynomials defined by $$H_n(\theta; q)= \sum\limits_{k=0}^n \frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}e^{i(n-2k)\theta}$$ are orthogonal in $\theta \in [0, \pi]$ with ...
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5answers
970 views

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 ...
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1answer
76 views

Orthogonality of functions related to Legendre polynomials

If $q\in P^{0}_{k}(I)$, i.e $q$ is a polynomial of degree $\leq k$ that vanishes at two end points of the interval $I=(0,1)$ and ...
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Finding general orthogonal polynomials

Many Special functions are orthogonal; for example, the sine and cosine function is an orthogonal function. Also, a couple of orthogonal polynomials are well-known. Now I'm asking the following: Given ...
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52 views

Big error in basis of tensor product space

Sorry I am currently somehow confused by the following: The Legendre polynomials $(P_l)$ form an ONB of $L^2(0,\pi)$ and the complex exponentials $(\frac{1}{\sqrt{2\pi}}e^{in \theta})_n$ form an ONB ...
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13 views

orthogonal polynomials: explicit representation

Consider a sequence of orthogonal polynomials $P_0(x) = 1$, $P_1(x) = x$, and recursively $P_{n}(x) = (a_n x + b_n) P_{n-1}(x) + c_n P_{n-2}(x)$ for some sequences of real constants $a_n$, $b_n$, ...
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Integrals of orthogonal polynomials and combinatorics

A beautiful result due to Evans and Gillis is that the function $$A(n_1,n_2,\cdots,n_r)=\int_0^\infty L_{n_1}(x)L_{n_2}(x)\cdots L_{n_r}(x)e^{-x}dx$$ counts the number of generalized derangements ...
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1answer
56 views

How to compute orthonormal polynomials from weight function?

I have a weight function $w(x)=e^{-x}$ with $x$ from $0$ (inclusive) to infinity. How would I compute the first four orthonormal polynomials with respect to this weight function?
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1answer
71 views

Explicit formula for the sum of an infinite series with Chebyshev's $U_k$ polynomials

$$\sum _{k=0}^{\infty } \frac{U_k(\cos (\text{k1}))}{k+1}=\frac{1}{2} i \csc (\text{k1}) \left(\log \left(1-e^{i \text{k1}}\right)-\log (i \sin (\text{k1})-\cos (\text{k1})+1)\right)$$ where $U_k$ is ...
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1answer
26 views

How Find all linear polynomials orthogonal to $f(t) = t$

$(f,g) = \sum_{k=0}^n f(\frac{k}{n})g(\frac{k}{n})$, where $f,g \in P_n$, the linear space of all polynomials of degree $\leq n$. If $f(t) = t$, find all linear polynomials $g$ orthogonal to $f$. I ...
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1answer
20 views

Jordan Normal Form of a Orthogonal Projection

I have the following question in my exam preparation: I have to: Find the minimal polynomial of T and the Jordan Normal Form of T. What I can't understand is ...
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71 views

Orthogonality in Space of Polynomials of Degree at Most 2

Let $E$ be the space of polynomials of degree at most $2$. On $E$ define $\langle f,g \rangle := f(-1)\overline{g(-1)}+f(0)\overline{g(0)}+f(1)\overline{g(1)}$ for $f,g \in E$. a). Show that this ...
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zeros of orthogonal polynomial satisfying $f_n (x) = (x-2n^2) f_{n-1}(x)- n^2 (n-1)^2 f_{n-2}(x)$

I am trying solving Romania TST 2014 problem #3 of day 3 http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3717806&sid=523a7da94c84b9ea1123a6b2f6302d34#p3717806 After finding out the ...
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0answers
23 views

Expected value of multivariate polynomial chaos

I'm a beginner in the use of polynomial chaos. If we have a univariate polynomial chaos : $Y=\underset{i}{\sum}\alpha_{i}\psi_{i}\left(X\right)$, thus : $\mathbb{E}\left(Y\right)=\alpha_{0}$ ...
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37 views

Is there a name for functions “opposite in nature” to orthogonal functions?

Suppose a function $f_n(x)$ is orthogonal over some domain $[a,b]$, then we have $$\left|\int_a^b f_n(x)f_m(x)dx\right| \left\{\begin{array}\\>0\text{ if }n=m\\ =0\text{ if }n\neq ...
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3answers
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Examples of orthogonal/orthonormal functions which are not finite degree polynomials?

I've been reading "Fourier Series & Orthogonal Polynomials" by Dunham Jackson. Great introductory read for anyone interested by the way! My question is, what are other examples of Orthogonal ...
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1answer
21 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 ...
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1answer
26 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 ...
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2answers
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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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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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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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32 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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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 ...
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1answer
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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) = ...
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1answer
31 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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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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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 ...
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1answer
49 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} ...
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1answer
50 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 ...
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2answers
96 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) = ...
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1answer
29 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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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 ...
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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 ...
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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 ...
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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 ...
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1answer
70 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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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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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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50 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 ...
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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 ...
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1answer
68 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
72 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
45 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.