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Tagged with chebyshev-polynomials orthogonal-polynomials
33
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Chebyshev polynomials orthogonal with respect to different weight function?
The following exercise appears in Ridgway Scott's Numerical Analysis:
Where $\omega_n(x)$ is the Chebyshev Polynomial of the first kind, that is $$\omega_{n+1}(x)=2^{-n}\cos((n+1)\cos^{-1}(x)$$
I ...
- 101
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Largest root of a linear combination of Chebyshev polynomials
I wonder if we can say something about roots of a linear combination of Chebyshev polynomials of the first kind. I have an example in my hand: $$(m+1)T_n(x)+(m-3)T_{n-2}(x)=0$$ for some $m>0$. I ...
- 433
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Density of Chebyshev nodes
While reading some notes, I came across the following statement:
``Chebyshev points have density $\mu(x) = \frac{N}{\pi\sqrt{1-x^2}}$".
I would like to understand where this formula comes from. ...
- 860
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why no initial conditions are required in the differential equation/eigenfunctions problem of orthogonal polynomials?
in section 4.2 of the book "Special functions, a graduate text" says the following:
We return to the three cases corresponding to the classical polynomials, with interval $I$, weight $w$, ...
- 105
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A stereographic projection for the Chebyshev polynomials
This question may be too vague for the MSE crowd, if so, please feel free to ask clarifying questions or just remove.
The Chebyshev polynomials are a family of orthogonal polynomials typically defined ...
- 533
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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)=...
- 7,880
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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 ...
- 927
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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 ...
- 7,880
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Do shifted Chebyshev polynomials form a complete set of independent functions?
Do Chebyshev polynomials form a complete set of independent functions? If yes, what can we say about their shifted versions? E.g. shifted Chebyshev polynomials of the first kind are defined as
$$T_{n}^...
- 7,880
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Show that first order Chebyshev polynomials are in fact polynomials
Given the 1st order Chebyshev polynomials
$$
G(x,t) = \sum_{n=0}^{+\infty} T_n(x) t^n = \frac{1-tx}{1-2xt+t^2}
$$
I'm wondering how can I show that $T_n(x)$ are polynomials ?
- 607
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Chebyshev in $n$ dimensions
The Chebyshev monomial integrals (of first and second kind) are
$$
I_k = \int_{-1}^1 x^k (1-x^2)^{\mp 1/2} \,dx
$$
Is anything known about their $n$-dimensional generalizations
$$
I_{k_1,\dots,k_n} = \...
- 1,570
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Chebyshev Polynomials of the first kind
If I write the Chebyshev polynomial of the first kind like this:
$T_n(x)=\cos(n\cos^{-1}x)$ for $x$ in $[-1,1]$.
It is clear that if $x=1$ then:
$T_n(1)=\cos(n\cos^{-1}1)=\cos(0)=1$ for all $n$, ...
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1
answer
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Higher-order Derivatives of Legendre Polynomials at endpoints
I am looking for simple formulas for higher-order derivatives of Legendre polynomials $P^{(k)}_{n}(\pm 1)$. For the Chebyshev polynomials, there is a simple formula
$$
\left. \frac{d^{p}T_n}{dx^p}\...
- 4,912
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How to find norm $||U_n||$ of Chebyshev polynomials of the second kind?
how to find norm $||U_n||$ and the values $U_n(\pm)$ of the Chebyshev polynomials of the second kind?
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Chebyshev Polynomials of the Second Kind from Orthogonality
I am tasked with finding the degree 5 Chebyshev-II polynomial, using the fact that it's orthogonal to those preceding it w.r.t the Chebyshev-II inner product. I am told to use the normalisation that ...
