Questions tagged [orthogonal-polynomials]
Questions pertaining to certain sets of polynomials that satisfy an orthogonality criterion with respect to some specified inner product.
738
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Convergence rate of Laguerre coefficients for polynomially bounded functions
Suppose $f:[0,\infty)\rightarrow\mathbb{R}$ satisfies:
$$f(x)= \sum_{n=0}^\infty \hat{f}_n L_n(x),$$
for some $\hat{f}_0,\hat{f}_1,\dots\in\mathbb{R}$, where $L_n$ is the $n$th Laguerre polynomial for ...
- 685
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2
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53
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Integral of a Generalized Laguerre Polynomial [closed]
I am looking for the solutions to the following integral:
$$
I_{n} =
\int_{0}^{\infty}x^{4}
\operatorname{L}_{n}^{3}\left(x\right)
{\rm e}^{-\left(n + 3\right)x/2}\,{\rm d}x,\qquad n \in\mathbb{N}_{0}
...
- 19
-1
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0
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40
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Uniquness of the orthogonality measure for generalized Laguerre polynomials [closed]
Let $\alpha>-1$. It is well-known that the measure $d\mu:=x^{\alpha}e^{-x}$ is the unique positive measure on $\mathbb{R}$ making generalized Laguerre polynomials $(L_n^{\alpha}(x))_n$ into an ...
- 107
1
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0
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28
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Concept of signal size for energy saving in optimization
I've been trying to redo this optimization problem from this paper, but on GEKKO Python code instead of MatLAB as they did, which is about finding the maximum Biodiesel concentration at final time:
J =...
- 19
1
vote
1
answer
85
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Question on orthogonal polynomials
I want to prove the Christoffel-Darboux formula, saying that for any three consecutive orthogonal polynomials $p_i$, we have
$$p_n(x) = (a_n x + b_n) p_{n-1}(x) - c_n p_{n-2}(x)$$
where $a, b, c$ are ...
- 1,285
1
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0
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50
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Polynomial Approximation of Piecewise Continuous Functions
I'm looking for results about constructing polynomial approximations of piecewise continuous functions. Specifically, I'm wondering about whether there is a straightforward approach to the following ...
- 11
2
votes
1
answer
79
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A quadrature rule given for this $\int_0^{\infty} e^{-t} f(t) dt = a_1 f(0) + a_2 f'(0) + a_3 f(t_3) + a_4 f(t_4) + R(f)$ using Gauss-Laguerre? [closed]
Can a Laguerre polynomial be used for this problem? How does $f'(0)$ square in?
Find coefficient and nodes for the following quadrature formula.
$\int_0^{\infty} e^{-t} f(t) dt = a_1 f(0) + a_2 f'(0) +...
- 23
1
vote
0
answers
51
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Recurrence relation for orthogonal polynomials with power law weight function
The objective is to find the recurrence relation for orthogonal polynomials with respect to the scalar product:
$$\langle f,g \rangle = \int_a^b f(x) g(x) w(x) dx,$$
where $0<a<b<\infty$, $w(...
- 21
1
vote
1
answer
42
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Weight function given polynomial basis
Consider the following polynomial inner product:
$$ (p,q)=\int_{-1}^{1} p(x) g(x) w(x) dx $$
It is well known that the polynomials of an orthogonal basis have simple roots, regardless of the weight ...
- 516
0
votes
1
answer
34
views
Why the orthogonal projection give minimal?
Given the data $ \vec{x}=(-2,-1,1,2) )$ and $( y=(1,1,-1,1) )$. Use an orthogonal projection to determine the coefficients $( a_{0}, a_{1}, a_{2} )$ of the quadratic polynomial function
$
\begin{...
- 11
0
votes
1
answer
42
views
An explicit formula for orthogonal functions
I am interested in orthogonal functions for the inner product
$$\int_a^b f(x)g(x) \alpha(x) dx$$
where $\alpha$ is a non-negative function.
Given linearly independant functions $f_0, \ldots, f_\ell$, ...
- 567
2
votes
1
answer
77
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Proof that monic polynomials $\{p_i(x)\}_{i = 0}^\infty$ each have $i$ distinct real roots in $[a, b]$
Problem: Let $\{p_i(x)\}_{i = 0}^\infty$ be orthogonal polynomials in the interval $[a, b]$ with respect to the inner product $(p(x), q(x)) \equiv \int_a^b p(x)q(x) \ dx$, where $p_i(x)$ is a monic ...
- 361
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0
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20
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Proof of completeness relation of complex exponetials
In fourier series I came across this completeness relation:
$$ \frac{1}{L} \sum_{n = -\infty}^{\infty} e^{-2\pi ni(x-x')/L} = \delta(x-x') $$
So I reduced this problem to proving:
$$ \sum_{n = -\...
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0
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65
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Hermite Polynomial and its Expectation
Currently, I'm stuck to some statement in a paper (in chapter 8: Nonlinear Model, from page 26 ~27). Although this topic generally covers statistics and machine learning theory, my main question is ...
- 769
1
vote
1
answer
53
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Lagrange interpolation and orthogonal polynomials
Suppose that $\{p_i(x)\}_{i=0}^{n}$ are pairwise orthogonal polynomials on the interval $[a,b]$, It means,
$$
\int_{a}^{b} p_i(x)p_j(x)dx = 0\ , \;{i\neq j}
$$
wherein $p_i(x)$ for all $i$ is a ...
- 188