# 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 ...
1 vote
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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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### 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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1 vote
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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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### 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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1 vote
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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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### 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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### 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 ...
1 vote
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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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There exists a way to expand a gaussian function into a series of Hermite polynomials as $$\sum_{n=0}^{\infty}r^{n}\left[H_{n}(x)\right]^{2} = \frac{1}{\sqrt{\pi(1-r^{2})}}\exp\left(\frac{2r}{1+r}x^{... 0 votes 1 answer 120 views ### Orthonormal polynomial basis of L^2([0,1]) I was wondering if, given a natural number i\in \mathbb N, there exists an orthonormal basis (w.r.t. the standard scalar product) (p_n)_{n \in \mathbb N} of L^2([0,1]) such that p_n is a ... 0 votes 1 answer 42 views ### Generating function of Meixner polynomial how can I prove that$$\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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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 ...