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

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Cubic function: Cardano's method

(Wikipedia link) So I am writing an essay on different ways to solve cubics. But I get stuck in the Cardano's method... Mainly is the part with Cardano's method's condition $\frac{q^2}{4} + ...
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1answer
39 views

Prove that $T_n$ satisfy $ \sum_{k=0}^{N-1}{T_i(x_k)T_j(x_k)} = \begin{cases} 0 &: i\ne j \\ l\neq 0 &: i=j \end{cases} \,\! $

The Chebyshev polynomials of the first kind satisfy the recurrence relation $$ \begin{cases} T_{n}(x)=2xT_{n-1}(x)-T_{n-2}(x) \qquad n \geq 2 \\ T_{0}(x)=1, \ \ T_{1}(x)=x \\ \end{cases} $$ The ...
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2answers
32 views

orthogonal polynomials and weight functions

how are orthogonal polynomials related to their weight function? is there an algebraic relationship other than the defining integral $$\int_a^b w(x)P_n(x)P_m(x)\,dx$$? thanks for the help!
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1answer
52 views

Orthonormal Basis of a function

An Orthonormal Family $\{e_k\}_{k\in\mathbb{N}}$ is a basis if and only if $$f=\sum^\infty_{n=1}\hat{f}(n)e_n \ \ \ \text{in} \ \mathcal{L}^2(\mathbb{R})$$ where $f\in\mathcal{L}^2(\mathbb{R})$ ...
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18 views

Why an orthonormal polynomial set over a continuous domain is not over a discrete one?

I would like to read the proof showing that a orthonormal polynomial set over a continuous domain is neither orthonormal nor complete over a discrete values on that domain. For example, Zernike ...
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1answer
40 views

Expressing associated Legendre polynomials in terms of unassociated Legendre polynomials

The associate Legendre equation is given as: $(1-x^2)\frac{d^2}{dx^2}y-2x\frac{d}{dx}y+\left[n(n+1)-\frac{m^2}{1-x^2}\right] y=0$ This becomes the standard unassociated Legendre equation for $m=0$. ...
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1answer
27 views

orthogonality of hermite polynomials when $x \rightarrow f(x)$

We know that if $m \ne n$: $$ \int_{-\infty}^{\infty} e^{-x^2}H_n(x)H_m(x)dx = 0 $$ And $$ \int_{-\infty}^{\infty} e^{-x^2} H_n(x)^2dx = \delta_{nm} 2^n n! \sqrt{\pi} $$ Now I am trying to find ...
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16 views

Inequalities for Laguerre polynomials

The following inequality holds, $$ \Big( 4\int_0^\infty rdr \big|\mathcal{L}_1(4r^2)\big|e^{-2r^2}\Big)^3 \geq 4\int_0^\infty rdr \big|\mathcal{L}_3(4r^2)\big|e^{-2r^2}, $$ where $\mathcal{L}_n(x)$ ...
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A “nice” orthogonal basis for translation invariant symmetric polynomials

It is going to be a rather long question, so I will first state it and then try to explain and motivate it. Take $\Lambda_n $ as the graded ring of symmetric polynomials of a field $F$ in $n$ ...
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30 views

Legendre polynomials and Dirac delta function

I have to show that $\sum_{l=0}^\infty \frac{2l+1}{4 \pi}P_l(cos\gamma)$; $\gamma = \angle ((\theta,\phi){,}(\theta ',\phi '))$, $P_l$ Legendre polynomials; vanishes if $(\theta,\phi) \neq (\theta ...
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2answers
80 views

Proving Bonnets' Recursion with Rodrigues' Formula

I would like to show that $(n+1)P_{n+1}(x)+nP_{n-1}(x)=(2n+1)xP_{n}(x)$ using Rodrigues' formula, not the generating function. I got to this point, but have not been able to progress further. ...
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316 views

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. ...
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25 views

Construct orthogonal polynomials from Gram matrix decomposition

Consider some bilinear form $$ B(u(x), v(x)) = \int_a^b u(x) v(x)\omega(x) dx $$ I would like to find a set of polynomials orthogonal with respect to thr form $B$, i.e $$ B(P_n(x), P_m(x)) = C_m ...
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28 views

How to find a quadrature formula of a specific shape?

What are the steps one needs to follow to find a quadrature formula of a certain shape with maximal degree of precision. For example: Find a quadrature formula of the following shape $\int_1^2 ...
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24 views

Approximation of Hermite functions

I'm looking for an "easy" proof of the asymptotic expansion of Hermite functions ($f_n(x)=H_n(x)e^{-x^2/2}$ where $H_n$ is the Hermite polynomials). The asymptotic expansion is $$ f_n(x) \sim_{n ...
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28 views

Estimation of the Hermite Polynomials using Plancherel-Rotach asymptotics

Suppose $H_n(x)$ is a Hermite Polynomial such that $$\int_{\mathbb{R}} H_n(x) H_m(x) e^{-x^2} dx = \delta_{m,n}.$$ I want to show for $ \phi_n(x) = H_n(x)e^{-\frac{X^2}{2}}$ $$ \left( ...
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1answer
27 views

Why are quadrature points given by the zeros of orthogonal polynomials?

We know there exists unique Gaussian quadrature formula. Its quadrature points are given by the zeros of the orthogonal polynomial. Why do we use only the zeros of the orthogonal polynomials in ...
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49 views

Laguerre theorem

I'm looking for a proof of the theorem 7, page 6, of this document : http://www.nipne.ro/rjp/2013_58_9-10/1428_1435.pdf Theorem 7 (E. Laguerre) Let $f \in \mathbb{R}[x]$ be a polynomial of degree ...
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38 views

Orthogonal polynomials

I was put on hold 2 times already for this question, I don't know how to solve it (If i knew how to solve it I wouldn't be bothering you ) and I don't know why it doesn't fit the rules of this site or ...
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1answer
86 views

Integrating Associated Legendre Polynomials

As part of a derivation for the question I asked here in Physics stackexchange, I am trying to calculate the following integral, but I am not sure how to proceed: ...
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47 views

Why are the Hermite Polynomials important?

I know a lot about the properties of the polynomials, but I don't know for what purpose they were developed or why they continue to be studies. Why are Orthogonal polynomials important besides their ...
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26 views

Othonormal basis for $L^2$ space on square

Can one describe an orthonormal basis for $L^2(\gamma,ds)$ where $\gamma$ is the square with vertices at $(1,1),(-1,1),(1,-1),(-1,-1)$ and $ds$ is the arc length. To be more precise can we express ...
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35 views

A cheap error estimate and a costly doubt

Carl de Boor poses the following problem in his A Practical Guide to Splines (1978 - Chapter II, p. 38, problem 4): The calculation of $||g|| = \max\{|g(x)| : a \le x \le b\}$ is a nontrivial ...
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2answers
106 views

show that nth Chebyshev polynomial is an nth order polynomial

Define the Chebyshev polynomial $T_n(x)=\cos(n\cos^{-1}(x)), n\geq 1, T_0=1)$. Show that $T_n(x)$ is an nth order polynomial This is my attempt, however I couldn't reduce it to a polynomial. ...
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1answer
23 views

Linear span of poisson kernels dense in $L^1(\mathbb{T})$

A paper I am reading ("Schur's Algorithm, Orthogonal Polynomials, and Convergence of Wall's Continued Fractions in $L^2(\mathbb{T})$" by Sergei Khrushchev...really a great paper) repeatedly mentions ...
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0answers
11 views

orthogonality of Lagrange polynomials with any associated orthogonal polynomials

I have very similar question with this link. Orthogonality of Lagrange Polynomials in Hermite Inner Product The problem is that I can not understand how can we use the fact that the associated ...
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51 views

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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32 views

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
57 views

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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30 views

Proof of identity with Hermite polynomials

Let's have Hermite polynomial, $H_{n} = e^{\frac{x^{2}}{2}}\frac{d^{n}}{dx^{n}}e^{-\frac{x^{2}}{2}}$. How to prove the identity $$ \tag 1 \sum_{n = 0}^{\infty}H_{n}(x)H_{n}(y)\frac{t^{n}}{n!} = (1 - ...
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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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14 views

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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0answers
49 views

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
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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 ...
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1answer
95 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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34 views

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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1answer
56 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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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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2answers
72 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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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
28 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
32 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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79 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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23 views

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
27 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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0answers
43 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
97 views

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 ...