All Questions
Tagged with chebyshev-polynomials linear-algebra
15
questions
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Formula for $T_k(x + 1)$ where $T_k$ is $k$-th Chebyshev polynomial of the first kind
Let $T_k(x)$ be the Chebyshev polynomial of the first kind of order $k$, i.e., the polynomial given by the recurrence relation
\begin{align}
T_0(x)=1, \quad T_1(x)=x, \quad T_k(x)=2xT_{k-1}(x)-T_{k-2}(...
- 1,450
2
votes
1
answer
626
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Chebyshev/Taylor polynomial error
So i have this equation:
$ y = x^2 * sqrt(1-x^2)$
when i work out the chebyshev polynomials and the taylor polynomials there is an offset of 0.18 between them.
to work out the polynomials i used:
$...
- 25
2
votes
1
answer
118
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Chebyshev Inequality - How is the following inferred ??
In chapter 3, Norm and Distance of Introduction to Applied Linear Algebra by Boyd, an example explaining the Chebyshev inequality for standard deviation is given as:
Consider a time series of return ...
- 23
1
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0
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458
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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 ...
3
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2
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338
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Is the following matrix defined by the roots of Chebyshev polynomial invertible?
Let $x_0, \dots , x_n$ be the roots of the Chebyshev polynomial $T_{n+1}(x)$.
We define:
$$A=\begin{pmatrix}
\frac{1}{\sqrt2}T_0(x_0) & \cdots & \frac{1}{\sqrt2}T_0(x_n) \\
T_1(x_0) & \...
- 301
0
votes
2
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477
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Proving that $T_0, T_1, T_2, ...$ are basis of $\mathbb{R}[x]$
I am given that the Chebyshev polynomial $T_n(x) \in \mathbb{Q}[x]$ is a polynomial such that $T_0(x) = 1$, $T_1(x) = x$ and for $n \ge 2$,
$T_n(x) = 2xT_{n-1}(x)-T_{n-2}(x)$
Now, I am supposed to ...
- 587
1
vote
1
answer
225
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Show that a matrix is nonsingular [closed]
What is an efficient way of showing that the matrix
$$\begin{align}
P\triangleq \begin{bmatrix}\cos\theta_1&\sin\theta_1&...&\cos\theta_n&\sin\theta_n\\
\cos2\theta_1&\sin2\...
- 899
1
vote
1
answer
1k
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Chebyshev polynomial
So I have a Chebyshev polynomial
where I am trying to prove that
$T_n(t)=\cos{(n \space \arccos{t})}, \space n=0,1,2...$ to form a system of orthogonal polynomials under the weighted inner product ...
- 456
0
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63
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Does $\sum\limits_{i=1}^{n} f(\lambda_i) = \text{tr}(f(A))$ hold for any function $f$?
We know that given a $n$-by-$n$ matrix $A$, and
its eigen values $\{\lambda_i\}_1^n$
its trace $\text{tr}(A)$.
then the following holds:
$$ \sum\limits_{i=1}^{n} \lambda_i = \text{tr}(A) $$
then ...
- 391
0
votes
1
answer
180
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A question about extending polynomial span to monomial basis
I have a final next week and our instructor gave us some examples with solutions but I could not understand some operations.
Inner product is
$$(p,q)=\int_{-1}^{1} p(t)q(t)dt$$
$W = Span\{1,t,t^2\}$ ...
- 37
2
votes
0
answers
99
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Constraints on a Chebyshev series representation of a CDF
My question is about deriving constraints for coefficients of a
Chebyshev series which represents a CDF.
Let $F(x)$ be the cumulative distribution function for $x\in [-1,1]$. Accordingly we know ...
- 499
7
votes
1
answer
1k
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Extending a Chebyshev-polynomial determinant identity
The following $n\times n$ determinant identity appears as eq. 19 on Mathworld's entry for the Chebyshev polynomials of the second kind:
$$U_n(x)=\det{A_n(x)}\equiv \begin{vmatrix}2 x& 1 & 0 ...
- 16.8k
1
vote
1
answer
251
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(dis)proving $Span((\cos nx)_{n\in\mathbb{N}})=Span((\cos^nx)_{n\in\mathbb{N}})$ in $\mathbb{R}^\mathbb{R}$
I am trying to show that $Span((\cos nx)_{n\in\mathbb{N}})=Span((\cos^nx)_{n\in\mathbb{N}})$ in $\mathbb{R}^\mathbb{R}$. ($0\in\mathbb{N}$)
I immediately thought of the Chebyshev polynomials :
$T_n(...
- 7,770
2
votes
0
answers
35
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Give bounds for degree of "decreasing" polynomial
Let $p$ be a polynomial of minimal degree to which the following is true:
$p(0) > \sum\limits_{i=1}^n \lvert p(i) \rvert + \sum\limits_{i=1}^m \lvert p(-i) \rvert$
Give upper and lower bounds for $...
- 165
3
votes
1
answer
444
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Question about chebyshev polynomial
Chebyshev polynomials are defined as such:
$$T_n(x)=\cos(n\arccos(x))$$
I'm asked to show that $\deg(T_j(x))=j$ and that $T_0,T_1,T_2,\ldots,T_n$ are an orthogonal basis of $\Bbb R_n[x]$.
I think I ...
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