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.

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124 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 ...
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127 views

Polynomials with specified ranges in intervals

Say I have two finite intervals $[a,b],[c,d]\subsetneq\Bbb R$ where $a<b<c-1<c<d$ and $b-a=d-c=s<1$. I want to find a polynomial $f \in \Bbb R[x]$ such that $$\forall x\in[a,b],\mbox{ }...
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223 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 ...
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75 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 & \ & \ &...
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100 views

Estimating Spline curve by OLS. Is a good idea to fix the knots at Chebyshev sites?

I am writing my master's degree thesis on a novel method for fixing knots in an adaptive way and while reading the literature I've found many references to the so-called Chebyshev sites. This sites or ...
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72 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 ...
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400 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 ...
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73 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}(\...
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195 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\...
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150 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-\...
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89 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. ...
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151 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 ...
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223 views

Why might one be inclined to think that polynomials of the form $\cos(n\arccos{x})$ would minimize error in Lagrange interpolation?

I was first introduced to Chebyshev polynomials (of the first kind) in the form $T_n(x)=\cos\left(n \operatorname{arccos}(x)\right)$. The usual recurrence relation was then derived from using trig ...
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56 views

Eigenvalues of Jacobian of Mandelbulb “triplex” power formula

I'm trying to find a lower bound for the distance estimate of the Mandelbulb fractal, or at least justify why using the scalar-derivative for distance estimation is so effective. The Mandelbulb ...
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64 views

One of Chebyshev's inequalities

How can I prove that this polynomial has at least n+1 zeroes? I have no idea.
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1answer
174 views

Change of Interval for Chebyshev–Gauss quadrature

I am curretly working to numerically evaluate an integral of the form: $$\int_{-1}^{1} f(x) \sqrt{1-x^2} dx$$ For this issue Gauss-Chebysehv integration of second kind seems ideal as it uses the ...
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53 views

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 ...
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51 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 $p(x)$: $...
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179 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 ...
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118 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 ...
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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) \...
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25 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 ...
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172 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. ...
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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 ...
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388 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 ...
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615 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&...
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144 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,\\ ...
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48 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) -...
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67 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 ...
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273 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_{...
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452 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 ...
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157 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 ...
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249 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}, ... , ...
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424 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), ...
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256 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 ...
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107 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 ...
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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 $...
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51 views

Show that the lowest-norm monic polynomial is of the form $\frac{(b-a)^n}{2^n}\frac{1}{2^n}T_n\left(\frac{2}{b-a}x-\frac{b+a}{b-a}\right)$

Let $T_n\in P_n[-1,1]$ the n-th Chebyshev polynomial. Show that the lowest-norm monic polynomial in $P_n[a,b]$ is of the form $$\frac{(b-a)^n}{2^n}\frac{1}{2^n}T_n\left(\frac{2}{b-a}x-\frac{b+a}{b-a}\...
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32 views

Chebyshev interpolating polynomial theorem

Could anyone explain this Chebyshev interpolating polynomial theorem 3.5 ?
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1answer
46 views

How would one go about computing a Chebyshev-type quadrature problem with general integration limits of [a,b] instead of [-1,1]?

More specifically, a problem of the form $\int_a^b\frac{f(x)}{\sqrt{1-x^2}}dx = \sum_{i=1}^{N}w_if(x_i)$, where $a,b \in [-1,1]$, $w_i$ are the weights, and $x_i$ are the abscissa. A quick search ...
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24 views

Chebyshev polynom problem.

Consider $P_n = \{Q_n(x) : \deg Q_n = n; \|{Q_n}\| = \max_{[a,b]}|Q_n(x)| = M >0\}$. Now consider $\bar{T}(x) = M T_n(\frac{2x - (b+a)}{b-a}) - $ Chebyshev polynomial (normed on space $P_n)$. We ...
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33 views

Chebyshev Approximation : concentrating nodes on specific intervals

I have a smooth function : $$\frac{1}{2\pi} \sin(2\pi x )~,$$ that I would like to approximate on an interval $[-a, b]$. However I already know that all the inputs are guarantied to be close to an ...
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72 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 ...
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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 $...
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165 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 ...
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67 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.
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41 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) & \...
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120 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 ...
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216 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 ...
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1answer
31 views

some problem about chebyshev series

Suppose that $f \in C[-1,1]$ has a chebyshev series $\sum_{n=1}^{\infty}a_nT_n$ (b) show that $E_n(T_{n+1})=1$ (c) show that $|E_n(f)-|a_{n+1}|| \le \sum_{k=n+2}^{\infty}|a_k|$ cf : $E_n(f)= \inf\{||...