Questions tagged [chebyshev-polynomials]

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively.

7
votes
0answers
91 views

Is there something like “associated” Chebyshev polynomials?

When I was experimenting with orthogonalization of polynomials $$p_n(x)=\begin{cases} 1-x^n&\text{if }n\equiv0\; (\operatorname{mod}2),\\ x-x^n&\text{otherwise}, \end{cases}$$ i.e. simplest ...
5
votes
0answers
107 views

How do you define the sample points used for Chebyshev approximation/interpolation?

It appears there are somewhat conflicting definitions of the points used in Chebyshev interpolation. Wikipedia and Numerical Recipes define the $x_j^{(n)}$ sample points for $(n-1)^\text{th}$-order ...
5
votes
0answers
65 views

What is this generalization of the Chebyshev polynomials?

For $\varepsilon>0$ consider the tridiagonal matrix $$L_{\varepsilon}=\begin{bmatrix} 0 & 1 & \ & \ & \ & \ & \ & \ \\ 1 & \varepsilon & 1 & \ & \ &...
4
votes
0answers
67 views

Probabilties using Central Limit Theorem

Let n be an independent random variables and the number of orders in a 120 minute period. Given that $\mu$ is 1.5 minutes and that $\sigma$ is 1 minute use the Central Limit Theorem to find the ...
4
votes
0answers
262 views

Complex Chebyshev Polynomials

Chebyshev Polynomials can be used to compute a very nearly minimax polynomial approximation of an analytic function on $[-1,1]$. Is there a complex analog that can compute a nearly minimax polynomial ...
3
votes
0answers
66 views

$\cos\frac\pi{n}$ Analytic expression

I recently found out that $$\sin\frac\pi5=\frac12\sqrt{\frac{5-\sqrt5}2}$$ Which means that $$\cos\frac\pi5=\frac{1+\sqrt5}4$$ I also recently found that if $n\in\Bbb N$, $$\sin nx=\sin x\,U_{n-1}(\...
3
votes
0answers
131 views

From polynomials to Chebyshev polynomials

I was wondering how they got from the polynomial to a Chebyshev polynomial as outlined here: In order to obtain “strong” stability, we replace the condition ($2.7$) by $$\left|\prod_{j=1}^N (1-\...
3
votes
0answers
86 views

Multivariate Appoximation

I have a mathematical model for a complex system which I would like to approximate it. My idea is to run this complex model once and produce some outputs, and then fit a polynomial for these outputs. ...
3
votes
0answers
118 views

Definite integral including the Chebyshev polynomial

I would like to know the proof of $$ \int_a^b \frac{T_n(x/a)T_n(x/b)\, dx}{x(b^2-x^2)^{1/2}(x^2-a^2)^{1/2}}=\frac{\pi}{2 ab}, 0<a<b, n \in \Bbb N $$ where $T_n(x)$ is the Chebyshev polynomial of ...
2
votes
0answers
28 views

Vieta's Formula for Chebyshev basis

Let $p(x)=x^d+\sum_{i=0}^{d-1} a_ix^i$. Then Vieta's formula tells us that the $a_i$ can be expressed as signed elementary symmetric polynomials of the roots $\{\alpha_1,\ldots,\alpha_d\}$ of $p(x)$: $...
2
votes
0answers
79 views

Chebyshev coefficients- interpolation on [a,b]

My problem is to solve a second order differential equation given two (Dirichlet) boundary conditions. $\frac{d^2y}{dx^2} = M/EI$ Both M and I are functions of x. Owing to complexity of the ...
2
votes
0answers
50 views

Minimal error chebyshev interpolation

Let's say the n-degree Chebyshev polynomials : $$ T_{n} (x)=\cos(n\arccos(x))$$ Make a polynomial such that: $$\mid y- P (x) \mid$$ be minimal, using the first three Chebyshev polynomials for the ...
2
votes
0answers
23 views

Numerical analysis: Chebyshev coefficient representation error.

I am unsure if numerical analysis questions are suitable for this forum, but I have nowhere else to ask, so if this question is inappropriate, tell me and I will delete it. If $x_k$ are the Chebyshev ...
2
votes
0answers
111 views

Pythagorean-like equation for generalized hyperbolic function

Trig functions satisfy $\cos^2t+\sin^2t=1$, which is an expression of the Pythagorean theorem. Hyperbolic trig functions satisfy $\cosh^2t-\sinh^2t=1$ which may perhaps be viewed as a generalization. ...
2
votes
0answers
76 views

Demystifying math: how could someone come up with Chebychev polynomials?

I hope this question is allowed, I am interested how you think someone could come up with the Chebychev polynomials, where I refer to them in the sense that someone would be interested in the ...
2
votes
0answers
247 views

Generating Chebyshev polynomials by Gram-Schmidt

Given the definition of Chebyshev polynomials in this form: $$T_n(x) = \cos(n\cos^{-1}x), n\ge 1, T_0=1$$ I want to show that using Gram-Schmidt procedure with set $\{1, x, x^2, \dots\}$ and weight ...
2
votes
0answers
431 views

How would you compute the eigenvalues of a finite difference operator?

Having read: What are eigenvalues of higher order finite differences matrices? I am still unclear how you would do this for an arbitrary matrix: $$A=\begin{pmatrix} a&b&e&0&0&0&...
2
votes
0answers
100 views

Find the cubic near-minimax or Chebyshev approximation for $f(x)=\sin(x)$

Find the cubic near-minimax or Chebyshev approximation for $f(x) = \sin(x)$ on the interval $[0,\frac{\pi}{2}]$. Attempt: The first four Chebyshev polynomials are \begin{align} T_0(x)&=1,\\ ...
2
votes
0answers
43 views

Solution to an $n^{th}$ order polynomial equation as a series in $n$

If I have an $n^{th}$ order polynomial set equal to zero, is there some way I can invert it to get a series expansion in terms of $n$? In particular I'm interested in solving the equation $$ U_n(x) -...
2
votes
0answers
62 views

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 ...
2
votes
0answers
342 views

Inner Product of Chebyshev polynomials of the second kind with $x$ as weighting

I have tried to solve the integral $${\int_0^1 U_n (x) U_m (x)x dx },$$ where ${U_n (x) }$ denotes Chebyshev polynomial of the second kind. Solving the integration and checking the result, I ...
2
votes
0answers
132 views

Size of coefficients of polynomials that satisfy a Chebyshev-like extremal property

The famous Chebyshev polynomials satisfy many extremal properties. One of these is that they attain the largest possible derivative over the interval [-1,1] among polynomials whose absolute value over ...
2
votes
0answers
196 views

How to find coefficients in a multivariate Chebyshev polynomial approximation

How do perform a multivariate Chebyshev approximation? Let \begin{align} \vec{x} & = x_{0}, x_{1}, ... , x_{n},\\ \vec{a} & = a_{0}, a_{1}, ... , a_{n},\\ \vec{b} & = b_{0}, b_{1}, ... , ...
2
votes
0answers
165 views

Why does this “incorrect” Chebyshev function approximation work better than the correct one?

I recently had the need to approximate this function $$f\left(x\right)=\begin{cases} \log\left(\frac{\pi+2\arcsin\left(x\right)}{\pi}\right), & x<0\\ -\log\left(2-\frac{\pi+2\arcsin\left(x\...
2
votes
0answers
326 views

Sampling a Chebyshev polynomial with the discrete cosine transform

I have a Chebyshev polynomial $f$ of degree $n$ in point-value form \begin{align} f&=:S = \left( \left( x_i, y_i \right) \right)_{i=0}^n, \tag{1} \\ x_i &= \cos\left( \frac{i \pi}{n} \right), ...
2
votes
0answers
213 views

Chebyshev polynomials approximation - Is there a way to generalize this

In an exam I was given this question: let $f(x)=x^3$. We want to find the best linear approximation (best in the sense that the maximal error is minimized) of $f$ in the interval $[-1,1]$ using ...
2
votes
0answers
95 views

how to prove this curious identity with the Chebyshev polinomials

we defined the Tm like this (where Tm are the Chebyshev polinomials) Then I showed this: And now I have no idea how to proove this: I also have to make the remark that I also proved that the ...
2
votes
0answers
31 views

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 ...
1
vote
0answers
29 views

Chebyshev series on the complex plane

Denote $$T_n(x) := \cos(n \arccos(x)),\,\, n\in \mathbb{N}$$ the Chebyshev polynomials. Let $f$ be a continuous function on $[-1, 1].$ It is well known that $f$ can be written in its Chebyshev series ...
1
vote
0answers
17 views

googd reference for studying second kind Chebyshev wavelets

Can anyone recommend a good reference for second kind Chebyshev wavelets? I want to know how can generate them by a mother wavelet and that how they can form a (orthonormal) basis for the space $...
1
vote
0answers
43 views

Chebyshev Polynomials: Properties of Derivatives

Show that: $T_n'(x)$=$2n\sum_{k=0\\k+n~~odd}^{n-1}\frac{1}{c_k}T_k(x)$ $T_n''(x)$=$\sum_{k=0\\k+n even}^{n-2}\frac{1}{c_k}n(n^2-k^2)T_k(x)$ where $c_0=2$ and $c_n=1$ for $n\geq1$ I tried using the ...
1
vote
0answers
65 views

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 ...
1
vote
0answers
33 views

Map Chebyshev nodes on an arbitrary shape?

Is it possible to map Chebyshev nodes on an arbitrary shape (e.g. in 2D: on a triangle, in 3D: on a cone or pyramid)? The Chebyshev nodes will be used as interpolation points? Thanks.
1
vote
0answers
26 views

Is the following matrix defined by the roots of Chebyshev polynomial invertible?

Let $x_0, \dots , x_n$ the roots of the Chebyshev polynomial, $T_{n+1}(x)$. We define: $\begin{pmatrix} \frac{1}{\sqrt2}T_0(x_0) & \cdots & \frac{1}{\sqrt2}T_0(x_0) \\ T_1(x_0) & \...
1
vote
0answers
32 views

Is there a general form of Chebyshev expansion coefficients for Gaussian distribution

$\newcommand{\chebyshevt}{\text{chebyshevt}}$ $\newcommand{\Norm}{\text{Norm}}$ I tried to calculate the coefficient for distribution $Norm(x, \mu, \sigma)$ via $$\int_{-1}^{1} \chebyshevt(x, t) \...
1
vote
0answers
86 views

Use 1-degree Chebyshev polynom to approximate $\cos(x)$ and calculate the error

The task is to give for $\cos(x)$ the nodes of the interpolation polynom of degree 1 that approximates the function on $[-\pi,\pi]$ the best as well as the related error. I want to solve this task ...
1
vote
0answers
113 views

Completeness relation for Chebyshev polynomials

The Chebyshev polynomials of the first kind $T_n(x)$ are known to form a complete orthogonal basis for functions on $[-1,1]$. I was looking for a proof of the completeness part, without any luck, when ...
1
vote
0answers
252 views

First kind Chebyshev polynomial to Monomials

Express First kind Chebyshev polynomial in terms of monomials First kind Chebyshev polynomial of order n ($T_n$) is defined in terms of cosine function as follow: 1) $T_n(\cos x)=\cos n x$ ...
1
vote
0answers
378 views

Explanation of numpy's Chebyshev curve fit

I'm writing a mini-library in C++ to find a 4th order Chebyshev polynomial (of the first kind) curve fit on set of x/y points varying in size (between 5-36 sets of points). I have found a pretty good ...
1
vote
0answers
50 views

In Chebyshev general solution, why the constants are set to ${0,1}$?

Chebyshev DE: $$(1-x^2)y''-xy'+n^2y=0$$ The general solution for the DE is: $$y(t)=C\cos(nt+\alpha)$$ I found that books sets the constants to be: ${C=1, \alpha=0}$ Why we set these values ? ...
1
vote
0answers
31 views

Prove: $ \|\hat{p}_n\|_\infty \leq \|p\|_\infty$ for all $p \in \mathbb{P}^a_n$

Let $a>1$ and $\mathbb{P}^a_n = \{p \in \mathbb{P}_n: p(a)=1 \}$. Define $\hat{p}_n \in \mathbb{P}^a_n$ by $ \hat{p}_n = \frac{T_n(x)}{T_n(a)}$, where $T_n$ is the Chebyshev polynomial of ...
1
vote
0answers
37 views

Proving a bound for the leading coefficient of a polynomial.

Show that every real polynomial $x\in C[a,b]$ of degree $n\ge 1$ with leading term $\beta_n t^n$ satisfies $$||x||\ge |\beta_n|\frac{(b-a)^n}{2^{2n-1}}.$$ I am having difficulty proving this. Here on ...
1
vote
0answers
183 views

Chebyshev polynomials increase more quickly than any other polynomial outside $[-1,1]$

In Appendix C3 of Shewchuk's excellent notes on conjugate gradient, it is stated without proof that Chebyshev polynomials... increase in magnitude more quickly outside the range $[-1,1]$ than any ...
1
vote
0answers
704 views

Any differences between Lagrange polynomial on Chebyshev points and Chebyshev polynomial?

I have to find an approximate continuous function that passes through a number of points. Many have said that the best (for my specific problem) is to use Chebyshev polynomial decomposition. I have ...
1
vote
0answers
57 views

Translation and multiplicative identities to Chebyshev polynomials

I am interested on finding translation and multiplicative identities for Chebyshev polynomials in the way they exist for Bernoulli ones: $$T_n (x+y) = ?$$ $$T_n (ax) = ?$$ Thanks
1
vote
0answers
90 views

Recurrence relation of Chebyshev nodes

Doing a few practice questions, I came across a proof based question. How would one go around to solve this? "Prove the following recurrence relation:" $\frac{d}{dx}T_{n+1}(x)=(n+1)T_n(x)+\frac{n+1}...
1
vote
0answers
244 views

Multivariate Clenshaw Chebyshev Algorithm (downward recursion)

I have recently written a code where I use Clenshaw's summation formula with Chebyshev polynomials $S(x)=\sum_{k=0}^nc_kT_k(x)=b_0+xb_1$ $T_{k+1}(x)=2xT_k(x)-T_{k-1}(x)$ $T_0(x)=1~~~ T_1(x)=x$ $b_{...
1
vote
0answers
437 views

Cubic Approximation to $e^x$ using Chebyshev Polynomial

Was trying to solve this: $C_r=\frac{2}{\pi}\int_{-1}^1\frac{e^xT_r(x)}{\sqrt{1-x^2}}dx$ where $r=0,1,2,3$ $T_r(x) =cosr[{cos}^{-1}x]$ While solving, I equated $x=cos\theta$ Therefore ...
1
vote
0answers
88 views

Is this polynomial irreducible over the rationals?

Prove (or disprove): Define $T_n(x)$ as the Chebyshev polynomial of the first kind with degree $n$. If $p$ is an odd prime, then $\sqrt{\frac{T_p(x)-1}{x-1}}$ is an irreducible polynomial over the ...
1
vote
0answers
99 views

Combinations of Chebyshev polynomials and sin functions

By chance, I see this formula $\int_0^1 T_{2n+1}(x)\sin(ax) { dx \over \sqrt{1-x^2}}=(-1)^n\frac{\pi}{2}J_{2n+1}(a)$ but what is the closed form if we have $\int_0^1 T_{2n}(x)\sin(ax) { dx \over \...