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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Gaussian Quadrature with Hermite Polynomials

I want to find a quadrature approximation of $\int f(x)e^{-\frac{x^2}{2}}$dx. Using the Hermite Polynomials 1, x, $x^2-1$, $x^3-3x$, $x^4 - 6x^2 + 3$. that is exact for polynomials up to order 7. Now ...
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orthogonal polynomials and determinant of jacobi matrix

In the book 'Orthogonal Polynomials of Several Variables' by Charles F. Dunkl and Yuan Xu page 11 is the picture below. I assume the matrix in the picture is related to Corollary 1.3.10 For the case ...
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Calculation of the integral of the Legendre polynomial of the second kind

Please tell me the possible options for calculating the integral of the form $\int\limits_{-\infty}^a\frac{Q_n(x)}{(x+b)^{n+2}}dx$, where $a\in[-2,-\infty)$; $b\in(-a,-\infty)$; $Q_n(x)$ - Legendre ...
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interlacing properties of zeros of orthogonal polynomials

Im working on the interlacing properties of zeros of orthogonal polynomials $p_n(x)$ - proved by Gabor Szegö (Orthogonal Polynomials - Theorem 3.3.2 ) The Theorem says: Let $x_1 < … <x_n$ be the ...
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Are there radial basis functions $\phi_n(x)$ which can be factored as $f_n(x) x^{\alpha}$ for $0<\alpha<1$ and $n>0$?

I am looking for a set of radial basis functions $\phi_n(x)$ on $\mathbb{R}^+=[0;\infty[$ which satisfy some othogonality condition $$ \int x \phi_n(x) \phi_{n'}^*(x) dx =\delta_{nn'}$$ and "...
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Show that for any monic polynomial of degree $n$, the inner product $\langle q,q \rangle \geq \langle p_n,p_n\rangle$

Let $\{ p_n \}$ be a family of monic orthogonal polynomials associated with a inner product $\langle f,g\rangle = \int_a^b w(x)f(x)\overline{g(x)}dx$. Show that for any monic polynomial of defree $n$, ...
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Prove that the Legendre polynomial recurrence relationship satisfies the defining differential equation

I am trying to show that from this recurrent relationship $$ (n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x) $$ that the Legendre polynomial $P_n(x)$ satisfies the differential equation $$ (1-x^2)P'' - ...
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  • 477
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How to prove the orthogonality of sequence of polynomials

my question is how one can go about proving the orthogonality of a sequence of polynomials. I know about the Strum-Liouville Theory (SLT). The situation in the SLT is that the coefficient of the ...
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Uncertainty propagation in non-samplig methods

I am able to "visualize" how uncertainty propagates from stochastic inputs in sampling methods (Monte Carlo simulation) since it is a consequence of repeated evaluations of the computational ...
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Generating Hermite polynomial with coefficient recurrance relation algorithm

I am writing a math paper for my numerical analysis class about orthogonal Hermite polynomials. I want to implement the algorithm for generating the "probabilist's Hermite polynomials": $$ \...
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Infinite sum of Laguerre polynomials: is $\sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^kL_n^k(x)^2=e^x+P(x)$, with $P$ a polynomial of degree $2n-1$?

I have the following expression: $$ \sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^k(L_n^k(x))^2, $$ where $$ L_n^k(x)=\sum_{j=0}^n(-1)^j\binom{n+k}{n-j}\frac{x^j}{j!} $$ is the usual associated Laguerre ...
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How $x^n$ is linearly represented by Legendre polynomials

I recently come across a problem with respect to Legendre polynomial as follows. Let $L^2[-1,1]$ be the Hilbert space of real valued square integrable functions on $[-1,1]$ equipped with the norm $\|f\...
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The values of $P_n(x)$ at the zeros of $P'_n(x)$

I recently come across a problem with respect to Legendre polynomial as follow. For any $n \in \mathbb{N}$, $P_n(x) := \frac{1}{2^n n!}\frac{{\rm d}^n (x^2-1)^n}{ {\rm d} x^n} $ is the classical ...
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Is it possible to find the value of a polynomial from the generating function?

Suppose I've to find $H_4(0),$ where $H$ represents the Hermite polynomial. I've only been provided with the following relation : $$e^{-t^2+2tx}=\sum_n H_n(x)\frac{t^n}{n!}$$ My first step is to ...
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Prove the Rodrigues formula by using the hypergeometric differential equation

Let us consider the classical orthogonal polynomials (COPs) $p_{n}(x)$, with weight $w(x)$. We have: $$\int_{a}^{b} w(x) p_{n}(x) p_{m}(x) d x= \begin{cases}0 & n \neq m \\ l \neq 0 & n=m\end{...
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Chebyshev equation for shifted Chebyshev polynomials of the first kind

The Chebyshev polynomials of the first kind ${\displaystyle T_{n}}(x)$ are given by the solutions of the following equation: $${\displaystyle (1-x^{2})y''-xy'+n^{2}y=0,} \quad \quad (1)$$ i.e. $y(x)=...
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Two orthogonal basis of polynomials with respect to a same inner product have the same roots

Let $V_n$ be the vector space generated by the set $\{1, x, \ldots, x^n\}$. We say that two polynomials $f(x)$ and $g(x)$ are orthogonal (with respect to a inner product) if $$\langle f(x), g(x) \...
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Show that the sum of Lagrange Polynomials $\sum_{i=0}^{n} L_{i}(t)=1 \quad \forall t \in R$ [duplicate]

I am reviewing a homework problem that is supposed to be really easy but I have trouble wrapping my head around it. For $j=0, \ldots, n \quad t_{j} \neq t_{i}$ if $ i \neq j $ we define the $n$ ...
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Spherical harmonics and dimension of SO(n-1) invariant subspace

Let $K \cong SO(n-1)$ the isotropy group for $e_n$. Let $H_m$ be space of homogeneous polynomials, and $Y_m$ the space of spherical harmonics. H_m^k denote the subspace of $K$ invariant polynomials in ...
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Proving operator is self-adjoint w.r.t. given inner product

Let $s$ be a nonnegative half-integer and $\mathscr P_s$ be the space of complex polynomials $p(z)$ of degree at most $2s$ in the formal variable $z \in \Bbb C$, equipped with the sesquilinear product ...
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Legendre polynomials from generating function

I have trouble to understand the following transform mentioned in Special functions and their applications by N. N. Lebedev, section 4.2 We can write the generating function of Legendre polynomials, $...
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Chebyshev polynomials semigroup property $T_n \circ T_m = T_{nm}$

Consider set of Chebyshev polynomials $T_n(x):\mathbb{R} \to \mathbb{R}$ given by formula $$ T_n(\cos(x)) = \cos(nx) $$ I am interested in elegant way to show that Chebyshev polynomials form a ...
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How to relate $\int_{-1}^{1}e^{ikru} P_{\ell}(u)du$ to $j_\ell(kr)$ in a simple way?

Consider the following epansion of the function $e^{ikru}$ in terms of Legendre polynomials, $P_\ell(u)$, $$e^{ikru}=\sum_{\ell=0}^{\infty}C_\ell(r)P_\ell(u)$$ where $k$ is a constant real parameter, ...
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orthogonality of Lagrange basis with legend nodes

I came across the following statement and I don't know how to justify it. If $L_i$ is a Lagrange basis, and $x$ is a zero of Legendre polynomial, then $$ \int_{-1}^{1} L_i(x)L_j(x) dx = \delta_{ij}w_j$...
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Construct First 3 Orthogonal Polynomials with Gram-Schmidt

Consider the set of functions $u(x)=x^n,\,\,$ with $n=0,1,2, \dots$. Use the Gram-Schmidt procedure to construct the first 3 orthogonal polynomials of: $$\text{Laguerre:} \;\;\;\;L_n(x),\;\;\;\; \text{...
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cubic polynomial for $L^2$ approximation.

Consider $f(x) = \sin \pi x$. Find the cubic polynomial giving $L^2$ approximation to $f$ on $[0,1]$. I suspect I need to find a polynomial of the form $p(x) = \sum\limits_{i=0}^3 c_ix^i$ and ...
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Term by term integration of orthogonal series

When dealing with an orthogonal expansion of a sufficiently smooth function $f$, i.e., $$ f(x)=\sum_{k=1}^\infty a_k p_k(x),\quad a_k=\int_a^b f(x) p_k(x) w(x)\, \mathrm{d}x, $$ what is the analogous ...
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How Can One Express u(xy) as a Diagonalized Transform Kernel, K(x,y)?

Consider a projection operator $P_{u}g(x)=<g(x),u(x)>$, where $u(x)$ is an eigenfunction normalized under an inner product, $<u_{m}(x),u_{n}(x)>=\delta_{m,n}$. (ASIDE: Inner products may ...
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Sturm-Liouville and Gram-Schmidt

Going back through some old class notes, I found this statement that I haven't been able to prove or determine if it's false and only the result of bad note-taking. Statement: Given arbitrary n-degree ...
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Three recurrence relations of orthogonal polynomial, monic and orthonormal polynomial

$\{p_n\}$ is orthogonal polynomial which satisfies $$p_{n+1} = (A_n x +B_n) p_n +D_n p_{n-1}$$ whose monic form $\tilde{p}_n$ satisfies $$x \tilde{p}_n(x) = \tilde{p}_{n+1}(x) + b_n \tilde{p}_n(x) +...
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What spherical functions can the N-th order limited spherical harmonics represent?

Background It is known that the spherical harmonics $Y_n^m$ with order $n$ and degree $m$ (such that $n \ge 0, -n \le m \le n$) are functions on the sphere which form a complete, orthogonal infinite ...
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Inner product of polynomials defined using determinant.

The following question was on my qualifying exam. I'd like to understand this question, but I've never seen anything like this, so I don't even know what key words to research or where to look for ...
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Solving $\delta(z) = \sum_\lambda A_\lambda e^{-\alpha z} \sin\big[\alpha R_\lambda (z-H)\big]$ for $A_\lambda$

The solution of an initial value problem I am working on becomes $$ \delta(z) = \sum_\lambda A_\lambda e^{-\alpha z} \sin\big[\alpha R_\lambda (z-H)\big]. $$ This equation is defined on the domain $0\...
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Gram- schmidt and chebychev polynomials

For all $n\ge 1$, let $(T_k)_{0\le k\le n}$ be the chebychev polynomials of degree $0 ,..,n$. I need to prove, using the dot product $(P\mid Q)=\int_{0}^{\pi} P(\cos t)Q(\cos t)\, \text{d}t$, that the ...
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Finite version of Mehler's Formula?

Mehler's formula is the following identity for Hermite polynomials $H_n(x)$: $$\sum_{n=0}^{\infty}\frac{t^n}{2^nn!}H_n(x)H_n(y)=\frac{1}{\sqrt{1-t^2}}\exp\Bigg(-\frac{t^2(x^2+y^2)-2txy}{1-t^2}\Bigg)$$ ...
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Do $\{T_{n}\left(\frac{x^2}{2}-1\right)\}_{n=1}^\infty$ form a basis for the even degree polynomials?

As we know, Chebyshev polynomials form a complete set of independent functions, i.e. they form a basis for the set of polynomials. Let us consider a class of shifted Chebyshev polynomials of the first ...
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Evaluating a 'shifted' orthogonal series of Hermite polynomials

In studying a physics problem, I've encountered the following series of Hermite polynomials $$ \sum_{n=0}^{\infty} \frac{H_n(x) H_n(y)}{n!2^n(n+1)^2}.$$ As of now, I am completely in a loss on whether ...
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Is there an expression for Hankel minors in terms of skew Schur polynomials?

There are known expression for Toeplitz minors in terms of skew Schur polynomials, see the paper entitled ''Toeplitz minors'' by Bump and Diaconis, or e.g. 1705.08067 and 1706.02574 In particular, ...
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On an orthogonal polynomial expansion of a function

I have some function that I am trying to study $ F(x) : [-1,1]\rightarrow \mathbb{R} $. I want write it in terms of a sum over Jacobi Polynomials $$ F(x) = \sum_n f^{\alpha,\beta}_n P^{\alpha, \beta}...
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Alternative orthogonality relations between associated Legendre polynomials

The usual orthogonality relations quoted for associated Legendre polynomials is: $$ \int_{-1}^{1}P_{n}^{m}(x)P_{n'}^{m}(x)dx=\frac{2(n+m)!}{(2n+1)(n-m)!}\delta_{n,n'} $$ However, I have come across ...
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Why does interpolating a function on an interval using orthogonal polynomials give the best possible approximation in that interval?

In the context of numerical integration, it is often said that quadrature methods which interpolate a function using orthogonal polynomials give the best possible approximation. I can understand why ...
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2 votes
1 answer
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Deriving Gauss-Hermite weights

I'm currently dealing with Gauss quadrature and I'm having trouble deriving the formula for the Gauss-Hermite quadrature weights. For reference: in my course the Hermite polynomials are defined with ...
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Jacobi Polynomials integral

I stumbled onto this integral: $$I=\int_{-1}^1 (1-x)^{\alpha-1} (1+x)^{\beta-1} P_m^{(\alpha,\beta)} (x)P_n^{(\alpha,\beta)} (x)\,dx$$ where $\alpha,\beta>1$. The Jacobi polynomials satisfy the ...
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Do Hermite polynomials satisfy $\int(H_n(x))^2 e^{-x^2} dx= 2n \int (H_{n-1}(x))^2 e^{-x^2} dx$?

I'm trying to prove that the norm of the Hermite polynomials (physicist's version) equals $ 2^n n! $. I stumbled upon this answer and I don't understand parts of the proof. First of all, it seems ...
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Rodrigues' Formula for Laguerre equation

This is exercise 12.1.2 a from Arfken's Mathematical Methods for Physicists 7th edition : Starting from the Laguerre ODE, $xy''+(1-x)y'+\lambda y =0 $, obtain the Rodrigues formula for its polynomial ...
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Gaussian Hermite expansion for complex numbers

There exists a way to expand a gaussian function into a series of Hermite polynomials as $$ \sum_{n=0}^{\infty}r^{n}\left[H_{n}(x)\right]^{2} = \frac{1}{\sqrt{\pi(1-r^{2})}}\exp\left(\frac{2r}{1+r}x^{...
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1 answer
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Orthonormal polynomial basis of $L^2([0,1])$

I was wondering if, given a natural number $i\in \mathbb N$, there exists an orthonormal basis (w.r.t. the standard scalar product) $(p_n)_{n \in \mathbb N}$ of $L^2([0,1])$ such that $p_n$ is a ...
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Generating function of Meixner polynomial

how can I prove that $$\sum_{n=0}^\infty{\frac{m_n\left(x;b,c\right)}{n!}t^n}=\left(1-t\right)^{-x-b}\left(1-\frac{t}{c}\right)^x$$ I tried using $$m_n\left(x;b,c\right)=\sum_{k=0}^{n}{\frac{n!}{(n-k)!...
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Can division of two generic polynomials be expressed as another polynomial?

The exact problem I am having: $C(x)=1+ c_1x$ and $D(x)=1 + d_1 x$, where $c_1, d_1 \ll 1$ but $c_1 x$ or $d_1 x$ can be comparable to unity for large values of $x$, so can't use binomial expansion ...
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3 votes
2 answers
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Derivation of Kravchuk polynomial identity

I am working my way through N.J.A. Sloane "An Introduction to Association Schemes and Coding Theory" and have got stuck proving the last of his identities for the Kravchuck (Krawtchouk) ...
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