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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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Can I form a complete set of functions using $e^{-nx}$?

If I start with the set of functions $e^{-nx}$ for all integers $n>1$, can I use them as basis to create a complete set of orthogonal functions on the interval $(0,+\infty)$? By complete I mean ...
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25 views

Let $S$ be the subspace of $\Bbb R^3$ spanned by the vector $x= (x_1,x_2,x_3)^T$ and $y= (y_1,y_2,y_3)^T$

(a) Let $S$ be the subspace of $\Bbb R^3$ Spanned by the vector $x= (x_1,x_2,x_3)^T$ and $y= (y_1,y_2,y_3)^T$, let $A =\begin{bmatrix}x_1 &x_2 & x_3 \\ y_1 &y_2 &...
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1answer
48 views

Can we refine this asymptotic for Laguerre polynomials?

I just found an interesting and useful limit for Laguerre polynomials: $$\lim_{n \to \infty} L_n \left( \frac{2r}{n+1/2} \right)=J_0(2 \sqrt{2r})$$ I'm using specifically this form of the argument ...
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10 views

Orthogonal polynomials on triangles: Connection with quadrature rules?

In a 1D interval $[a, b]$, quadrature rules and orthogonal polynomials are tightly interconnected. For example, given the $n$ roots $t_j$ of the $n$th orthogonal polynomial $p_n$, one has the ...
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18 views

Evaluating an integral using Gegenbauer polynomials

I want to evaluate the following integral $$\int \frac{(r-r'\cos\theta')^2r'^2\,dr'\sin\theta'\,d\theta'\,d\phi'}{(r^2+r'^2-2rr'\cos\theta')^{3/2}}$$ Working that a little bit i end up with this ...
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23 views

Orthogonality of generalized Newton symbol

Consider the functions $P_{n}(x)={x \choose n}.$ My question is, if there exists a measure $\mu$ with support being a subset of $(0,\infty)$ such that the family $\{P_{n}\}$ is orthogonal in $L^{2}(\...
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56 views

Define on $P3$ the inner product $<f,g>=\int_{-1}^1 f(t)g(t)dt$, find orthogonal projection

Define on $P3$ the inner product $\langle f,g \rangle=\int_{-1}^1 f(t)g(t)dt$. a) find the orthogonal projection of $p(x)=x^3$ onto $P2$ I know the orthogonal projection formula, but how do I solve ...
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57 views

What is this generalization of the Chebyshev polynomials?

For $\varepsilon>0$ consider the tridiagonal matrix $$L_{\varepsilon}=\begin{bmatrix} 0 & 1 & \ & \ & \ & \ & \ & \ \\ 1 & \varepsilon & 1 & \ & \ &...
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28 views

Integral involving the Associated Laguerre polynomials

I'm trying to solve this integral $\int_{0}^{\infty} L^n_p L^n_{p'} e^{-x} x^{n-1} dx = \dfrac{1}{n} \dfrac{(p!)^3}{(p-n)!} \delta_{pp'}$ I started with integration by parts where $u = $ $L^n_p L^...
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28 views

Projection on a subspace

An inner product is defined on $P_3$ such that $<f, g>$ $=$ $\int _{-1}^1\:f\left(t\right)g\left(t\right)dt$. What is the orthogonal projection of $p(x)$ $=$ $x^3$ onto the subspace $S$ $=$ $\...
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An closed-form expression of an integral of Chebyshev series and exponential function

Does the following integral has a closed-form expression? $\int_{-1}^{1} T_n(x)\exp(i\pi x)dx,$ where, $T_n$ is the Chebyshev polynomial of degree n.
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2answers
70 views

Orthogonal Projection on a Polynomial Space

An inner product is defined on $P3$ such that $<f, g>$ $=$ $\int _{-1}^1\:f\left(t\right)g\left(t\right)dt$. What is the orthogonal projection of $p(x)$ $=$ $x^3$ onto $P2$? So I got that $f_1\...
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Identifying the orthogonal polynomial from the recurrence relation

I am trying to see if this recursion relation can be solved in terms of standard orthogonal polynomials : $$ p_{n+1}(x)=2 x p_{n}(x)-2(n+1) p_{n-1}(x), $$ with $p_0(x)=1$ and $p_1(x)=2x$. These ...
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1answer
20 views

The generalized Laguerre polynomials: Are there any expressions valid for any case?

There are general expressions of the generalized Laguerre polynomials. For example: $$ L_n^{(\alpha)}(x) = {}_{n+\alpha}\mathrm{C}_n\ {}_1F_1(-n, \alpha+1, x), \hspace{20pt}(1) $$ $$ L_n^{(\alpha)}(x)...
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1answer
28 views

Use Gram-Schmidt process on Chebychev polynomials

I need help using the Gram-Schmidt orthogonalization process to derive the first four orthonormal Chebychev polynomials. Using the range $[-1,1]$ and the weight function $w(x)=(1-x^2)^\frac{1}{2}$. ...
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1answer
79 views

Orthogonal polynomials with respect to discrete probability distributions

The polynomials from the Askey scheme are orthogonal with respect to a standard probability distribution. For example, Hermite polynomials (in a suitable form) are orthogonal with respect to a ...
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8 views

Polynomials orthogonal wrt Rayleigh distribution

As per the title, have the class of polynomials that are orthogonal with respect to Rayleigh distribution been studied? Do they have a name?
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1answer
39 views

Algorithm for orthogonalizing polynomials with specific inner product

I am attempting to generate a as big as possible collection of orthogonal polynomials $p_1, p_2, ..., p_n$, $\left\langle p_i, q_i\right \rangle = \delta_{ij}$ where the inner product is with respect ...
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39 views

$L_2$ scalar product between Hermite polynomials

I am trying to compute the $L_2$ scalar product between (probabilists’) Hermite polynomials (defined as in Wiki) with Gaussian weight and different scales, i.e. for some constants $c, d$: $$\frac{1}{\...
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Orthogonal polynomials as eigenfunctions of a second-order difference operator

I am Reading the theorem 6.1.3 of this book https://books.google.com.mx/books?id=RusIDAAAQBAJ&pg=PA146&lpg=PA146&dq=up+to+normalization,+the+charlier,+krawtchouk,+meixner,+and+chebyshev%E2%...
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Weird Integral Involving Hermite Polynomials

I have stumbled upon the following integral involving the Hermite polynomials: $$ I(m) = \int_\mathbb{R} e^{i m x} \left[ e^{-\frac{x^2}{2}} H_m(x) \right] dx \, , \quad m \in \mathbb{N} \cup \{0\} \...
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1answer
37 views

An Addition formula for Hermite polynomials

My question concerns an addition formula that can be found on the Wikipedia page of Hermite Polynomials but I can't find it anywhere else. The well-known formula that can be found in many books is the ...
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1answer
35 views

Sum involving Hermite polynomials

I am wondering if there is a simpler form of the following summation involving the (physicists') Hermite polynomials: $$\sum_{k=0}^{n}\frac{H_k(x)}{(2i)^k},$$ where $i=\sqrt{-1}$ is the imaginary ...
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1answer
134 views

Find the Fourier-Bessel Series for $f(x)$ With Respect to the Orthogonal Set: How Was $w(x)$ Found?

I have the following problem: If $f(x) = x$, $0 < x < 2$, find the Fourier-Bessel series for $f(x)$ with respect to the orthogonal set $\{ J_1 (k_n x) \}$, where $k_n$ is the $n$th positive ...
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0answers
61 views

Integral of a triple product of Laguerre polynomials

I would like to know if there's an exact expression for this integral in terms of known elementary or special functions: $$\int_0^\infty \exp \left(-\frac{a+b+c}{2}x \right) L_j (a x) L_k (b x) ...
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1answer
52 views

How to prove two function $\phi_m$ and $\phi_n$ are orthogonal?

A while back I found a specific proof for the orthogonality of two functions but I can't seem to find it online anywhere. I just want to make sure the proof I've given is defined exactly, which is: "...
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The Distribution of Abscissae and Sums of Weights in Gaussian Quadrature

Let $n$ be a positive integer, and for $i = 1, 2, \ldots, n$, let $x_i$ be the $i^\text{th}$ abscissa for $n$-point Gaussian quadrature, and $w_i$ the associated weight, so that $-1 < x_1 < x_2 &...
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68 views

Express these polynomials in terms of orthogonal ones

In a problem in QM I faced with this polynomials, sadly they are not orthogonal. I was wondering if someone else knows these polynomials, I was looking up and didn't find anything about them but they ...
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0answers
23 views

Help finding source papers for questions concerning orthogonal polynomials.

I solved some problems concerning orthogonal polynomials from chapter four of the book "Classical and Quantum Orthogonal Polynomials in One Variable" by Mourad E. H. Ismail. I would like to find the ...
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I want to prove that $ \int_{-1}^{+1}xp_{s}(x)p_{r}(x)dx=\delta_{r,s+1}\frac{2r}{(2r+1)(2r-1)}+\delta_{s,r+1}\frac{2s}{(2s+1)(2s-1)}. $

Please make an illustration to me in proving of the following. Problem: Assume that $ p_{n}(x)$ is a Legendre polynomials. I want to prove that $$ \int_{-1}^{+1}xp_{s}(x)p_{r}(x)dx=\delta_{r,...
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1answer
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Determining Rodriguez formula for Legendre polynomials

While proving Rodriguez formula for Legendre classical orthogonal polynomials I found a part I cannot prove. Namely, if we define $Q_n(x) = \frac{(-1)^n}{2^n n!}((1-x^2)^n)^{(n)}$ and further observe $...
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15 views

Is there a Grassman equivalent of orthogonal functions?

Orthogonal functions such as the Hermite functions work with commutative variables. Is there a similar thing that works with Grassman variables?
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1answer
83 views

Use Gram-Schmidt orthogonalisation to orthogonalise the system of vectors

I have been working on this problem, we are given the below system of vectors $f_{1} = x, f_{2} = \cos(x), f_{3}= \sin(x)$ from the inner product of $C_{\mathbb{R}}[-1,1]$ and we have to ...
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64 views

How to construct an orthogonal, complete set of functions on the interval $[0, L]$ with given conditions at $x=0,L$?

The functions must be at least $C^2$, preferably $C^\infty$. Completeness for square-integrable functions is sufficient. The conditions at the points $x=0,L$ prescribe the values of the function and ...
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0answers
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Is there a name for “almost-subspaces” $U$ that do not satisfy $cu \in U, \forall u \in U$? If so do they have any properties?

I am working with a 'near-subspace' in which the subspace is defined by $$ U := \{p(x) \in \mathbb{P}^3 \mid p(x)> 0, 0\le x \le 1 \lor p(x) \equiv 0 \} $$ This satisfies that $0\in U$ and it ...
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0answers
39 views

Is any algebraic number a root of a given orthogonal polynomial?

Let $S=\left\{P_n(x)\right\}_{n=0}^{\infty}$ be any sequence of $n$-degree polynomials which are orthogonal in the interval $(a,b)\in\mathbb{R}$. If $\xi$ is an arbitrary algebraic number such that $a&...
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1answer
46 views

Basis functions for a Galerkin procedure

For a Galerkin procedure, I am trying to construct a set of linerly independent functions $\{\varphi_n\}_{n = 1}^N$ satisfying $$ \varphi_n(0) = 0, ~~ \varphi_n'(1) = 0, $$ for all $n \geq 1$. A ...
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1answer
72 views

If three functions A, B and C are mutually orthogonal, is the inner product of all three functions equal to zero?

Let A, B and C be mutually orthogonal functions. If so, would the following equation hold true? Is there a theorem proving this? $\int_a^{b}ABCdx = 0$
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Deriving a new set of complete orthogonal basis functions inside an interval?

I am quite a noob in this area, so bear with me if my question doesn't make sense, and be kind enough to let me know why it is so. I am interested in deriving a new set of orthogonal basis functions (...
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0answers
28 views

Are there any complete orthogonal basis functions inside unit sphere?

I went through so much literature, but couldn't find any orthogonal complete functions within the boundary $0 < r <1, 0 < \theta <\pi, 0 < \phi <2\pi$ other than 3D Zernike ...
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0answers
60 views

complete orthogonal basis functions within unit sphere

I know that spherical harmonics are complete on the surface of the unit sphere, and 3D Zernike Polynomials are complete inside the unit sphere. Are there a set of orthogonal complete basis functions ...
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0answers
70 views

Complex generating function for Legendre polynomials

I want to clarify, does the series $$ \sum_{n=0}^{\infty} P_n(x) e^{int} \qquad (1) $$ converge? I know that for all complex $z$, such that $|z|<1$, we have formula $$ \sum_{n=0}^{\infty} P_n(x) z^...
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0answers
44 views

Eigenvalues and orthogonal function expansions

Suppose I have a function $f: [-1,1] \rightarrow \mathbb{R}$ expanded in terms of Legendre polynomials, $$ f(x) = \sum_{n=0}^{\infty} a_n P_n(x) $$ and I evaluate this function at a large number of ...
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0answers
30 views

Problem about Integration of several orthogonal polynomials

In the literature, there are several orthogonal polynomials, like Hermite Polynomials, Legendre Polynomials. However, I would like to ask whether there are any explicit formulas for the integration of ...
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0answers
27 views

Inner product function

I need to calculate the inner product to approximate function in Hilbert space. I need to use expression $\lambda_k=\frac{(f \cdot g_k)}{(g_k \cdot g_k)\cdot(f \cdot g_k)}$ =$\int_a^bf(x)g_k(x)w(x)...
3
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1answer
87 views

Precise conditions on Sturm-Liouville Theorems

In Sturm-Liouville (SL) theory (https://en.wikipedia.org/wiki/Sturm-Liouville_theory), there are three fundamental theorems concerning the solutions of the SL differential equation, $ \frac{\mathrm{d}...
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0answers
20 views

Generalization of monomial orthogonality?

In this answer a simple ad-hoc inner product for monomials is defined, since they are known to be orthogonal: if $P = \sum_{n \ge 0} a_n X^n$ and $Q = \sum_{n \ge 0} b_n X^n$, then $$\left\langle ...
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0answers
90 views

Orthogonality Condition for hypergeometric functions

I would like to learn if there is an orthogonality relation for the hypergeometric functions, namely if I can calculate the integral of the product of two Gaussian Hypergeometric functions $_2F_1$ ...
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2answers
100 views

Showing that the functions $\sin(2n+1)x$ are orthogonal with respect to the inner product $\int_0^{\pi/2}f(x)g(x)dx$

I couldn't find this anywhere here, so I'll ask it. How would I show: $$\{\;\sin((2n+1)x)\;\}$$ on $[0,\pi/2]$, where $n>0$ and is an integer, is an orthogonal set with respect to the inner ...
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0answers
88 views

Definite integration of 2D Legendre Polynomials

Let a 2D Legendre Polynomial be defined as a product of 1D Legendre polynomials, $V_n^m(x,y) = P_n(x)P_m(y)$. I want to evaluate the integral, $ \int_{x=-1}^1 \int_{y=-\sqrt{1-x^2}}^{\sqrt{1-x^2}} ...