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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show that nth Chebyshev polynomial is an nth order polynomial

Define the Chebyshev polynomial $T_n(x)=\cos(n\cos^{-1}(x)), n\geq 1, T_0=1)$. Show that $T_n(x)$ is an nth order polynomial This is my attempt, however I couldn't reduce it to a polynomial. \...
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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 \...
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What is the connection between Taylor series and Chebyshev polynomials?

Can somebody help me find some historical references for the connection between Chebyshev polynomials and the Taylor series for sine and cosine functions? We know that Chebyshev polynomials are used ...
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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 ...
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Integral calculation/transformation $\varphi_n= 2x\varphi_n$

Please help with below integral transformation. \begin{align*} T_n(x) & = \cos(n\cos^{-1} x)\\ w(t) & = (1 - x^2)^{-\frac{1}{2}}\\ \varphi_n(x) & = \int_{-1}^{1} \frac{T_n(t) - T_n(x)}{t -...
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What to do when the chebyshev point is equal to data point in lagrange interpolation?

I am going to use Lagrange interpolation using Chebyshev nodes using the following formula $$\sum_x \prod_{k=0,k\not={j}}^n \frac {x-y_k}{y_j-y_k} f(x) $$ in which $x$ in my data points, $y_k $s ...
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Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials. Pointwise Lagrange ...
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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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dividing a chebyshev polynomial by another polynomial

If I took a Chebyshev polynomial, is it possible to divide it completely by something that isn't a chebyshev polynomial? edit - the question was answered but people were not sure about what I was ...
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polynomial approximation - basic chebyshev question

I was asked to find the best linear approximation to $f(x)=x^2$ in $x \in [0,1]$ using chebyshev polynomials, meaning, using the known property that $2^{1-n}T_n(x)$ is the best approximation to $0$ at ...
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How to improve stability of numerical solutions to partial differential equations

This is a quite general question, but I am working with a system of partial differential equations in two variables. There is one time direction $t$ and one spatial direction $z$ and the numerical ...
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Limit of Ratio of Chebyshev Polynomials

I have been trying to compute the limit $$\lim_{n\to\infty}{{U_n(x)^2}\over{U_{n-1}(x)^2+U_n(x)^2}}$$ where $U_n(x)$ is the $n$-th Chebyshev polynomial of the second kind and $x\ge 1$. Using software ...
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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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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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How do I solve for the zeros of a Chebyshev polynomical? (on a computer)

I am working on a computer program and have a method that returns a number for a given $x$, $y$. So $f(x, y) = z$, where $f$ is my method. if I know $y$ and $z$, can I find what $x$ will be, without ...
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Does this formula holds true for $|x|>1$?

When the first two Chebyshev polynomials $T₀(x)$ and $T₁(x)$ are known, all other polynomials $T_{n}(x),n≥2$ can be obtained by means of the recurrence formula $$T_{n+1}(x)=2xT_{n}(x)-T_{n-1}(x)$$ My ...
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Multiplication of polynomials in Chebyshev basis

For polynomials in the monomial basis like $p_n(x) = \sum_{k=0}^N a_k x^k $, the product of 2 polynomials is can be either found though the convolution of the 2 corresponding polynomial vectors or ...
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Power (monomial) form conversion to Chebyshev form

Given a polynomial in the monomial form e.g. like $p(x) = a_0 + a_1 x + \ldots + a_{n-1} x^{n-1} + a_n x^n$, how is it possible to convert it to the Chebyshev basis (i.e. represent it as a linear ...
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Chebyshev's Theorem regarding real polynomials: Why do only the Chebyshev polynomials achieve equality in this inequality?

In the book Proofs from The Book by Aigner and Ziegler there is a proof of 'Chebyshev's Theorem' which states that if $p(x)$ is a real polynomial of degree n with leading coefficient $1$ then $$ \...
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Chebyshev Polynomial Recurrence relation

the problem states: "Show that $\cos(n\theta)$ is a polynomial in $\cos(\theta).$" Now, using De Moivre's and Binomial theorems i get that $$\cos(n\theta) = \sum_{k = 0, evens}^{n}\binom{n}{k}\cos^{n-...
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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(...
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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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Existence of polynomial such that $P_n(cos\theta)=cos(n\theta)$

Is there a way of proving existence of a polynomial $P_n(x)$ such that $\cos{(n\theta)}=P_n(\cos{\theta})$ without knowing the Chebyshev polynomials a priori?
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Is it possible to calculate the roots of the difference between successive terms of this polynomial series $\rm{P}_n (x)=x\rm{P}_{n-1}-r\rm{P}_{n-2}$

Consider the polynomial series defined by the following recursion formula: $$ \begin{align} &\mathrm{P}_0 = 1 \\ &\mathrm{P}_1 = x-r \\ &\mathrm{P}_n = x\mathrm{P}_{n-1} - r\mathrm{P}_{n-...
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Product of Chebyshev polynomials of the second kind?

So Wikipedia has this formula for a product of two Chebyshev polynomials of the second kind evaluated at a fixed $x$ with different indices: $$ U_n(x)U_m(x)=\sum_{k=o}^{n}U_{m-n+2k}(x) $$ Which would ...
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What is a tensor-product Chebyshev grid?

What is the difference between "Chebyshev grid" and "tensor-product Chebyshev grid"? Are they defined on a 2D vector?
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How to find Chebyshev nodes

I want to use Chebyshev interpolation. But I am a little confused for finding Chebyshev nodes. I use the following figure to illustrate my problem. Consider I have a vector of numbers I depicted as a ...
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When does $|P_n(x)|=1$ hold?

Let $P_n = \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{2k} (X^2-1)^k X^{n-2k}$ be the $n$-th Tchebyshev polynomial. I want to solve $|P_n(x)|=1$ for $x\in [-1,1]$ I checked that $\large\cos\frac{k \pi}...
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Link between Chebyshev polynomials and best approximants

I'm reading Interpolation and Approximation by Davis, more specifically "Best Approximation" Chapter VII. Let $n \in \mathbb N$. Let $C[a,b]$ denote the set of continuous real functions over $[...
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Use the zeroes of T3 to construct an interpolating polynomial

Use the zeroes of T3 to construct an interpolating polynomial of degree two for the function x^3 on the interval [-1,1] Okay, so I have been looking at Finding the zeroes using Chebyshev polynomials ...
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Numerical evaluation of polynomials in Chebyshev basis

I have high order (15 and higher) polynomials defined in Chebyshev basis and need to evaluate them (for plotting) on some intervals inside the canonical interval $[1,\,-1]$. A good accuracy near 1 and ...
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chebyshev nodes on a 2D grid

I want to use chebyshev nodes for interpolation using lagrange formula. My grid is two dimensional and i do not know how to determine the nodes of chebyshev in a 2D grid point?
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Is there a smooth map from the square to the deltoid?

Is there a $C^\infty$ map between a unit square in $\mathbb R^2$ and a deltoid like this one The deltoid is obtained by varying the angles $\theta_1$, $\theta_2$ in the equations \begin{align} x_2 &...
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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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How to solve this integral manually

I was very much surprised that the Wolfram Online Integrator solved this integral very readably and in an elegant way : $${\large\int}\frac{\cos\left(\left(11+\frac 12\right)x\right)}{\cos\left(\left(...
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Strange property of parametrization of a class of plane curves

My studies lead me to the following parametrization of perhaps a new class of plane curves ( which are similar in shape to the classical sinusoidal spirals but not identical ). If the curves are not ...
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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 ...
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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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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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Chebyshev coefficients from a polynomial

Is there an efficient algorithm for finding the coefficients in a Chebyshev basis of a polynomial? That is, given the set of $a_k$ such that: $p_n(x) = \sum_{k=0}^N a_k x^k $ Find the set of $c_k$ ...
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Evaluating sinusoid at Chebyshev points

Suppose I have a sinusoid $f(t) = A \cos(\omega t + \theta)$ and I want to evaluate it at Chebyshev points of the second kind ($\cos(\frac{2 \pi i}{N}), 0 \le i \le N, i \in \mathbb{Z}$), and then ...
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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 $T_0,T_1,T_2,...,T_n$ are an orthogonal basis of $\mathbb R_n[x]$. I think I can ...
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How to get from Chebyshev to Ihara?

I have competing answers on my question about "Returning Paths on Cubic Graphs Without Backtracking". Assuming Chris is right the following should work. Up to one thing: The number of returning paths ...
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Combinatorial Interpretation of Graph Theoretical Relation Involving Chebyshev Polynomials

Given a graph $G$ and its adjacency matrix $A$. The $(i,j)$-th element of $A^r$ gives the number of ways to get from vertex $i$ to $j$ in $r$ steps (including backtracking). Now, the number of ...
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Coefficients of Chebychev Polynomials

Is there a known formula for the coefficient of x^k in the nth chebychev polynomial of the first kind?
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Prove $T_n(x)$ of Chebyshev Polynomial given the recurrence relation

Using the recursion formula for Chebyshev polynomials, show that $T_n(x)$ can be written as $$T_n(x)=2^{n-1}(x-x_1)(x-x_2)...(x-x_n)$$ where $x_i$ are the $n$ roots of $T_n$ The recurrence relation:...
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Calculation of Chebyshev coefficients

The Chebyshev polynomials can be defined recursively as: $T_0(x)=1$; $T_1(x)=x$; $T_{n+1}(x)=2xT_n(x) + T_{n-1}(x)$ The coefficients of these polynomails for a function, $\space f(x)$, under ...
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Trigonometric Identities for $\sin nx$ and $\cos nx $

These are generalizations of simple trigonometric identities for $\sin 2x$ and $\cos 2x$, but in general how can we prove them? $$\sin nx =\sum_{k=1}^{\left\lceil\frac{n}{2}\right\rceil}(-1)^{k-1}\...
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How to best approximate higher-degree polynomial in space of lower-degree polynomials?

My question is: Find the best 1-degree approximating polynomial of $f(x)=2x^3+x^2+2x-1$ on $[-1,1]$ in the uniform norm(NOT in the least square sense please)? Orginially, as the title of the post ...
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Integrating Chebyshev polynomial of the first kind

I'm trying to evaluate the integral of the Chebyshev polynomials of the first kind on the interval $-1 \leq x \leq 1 $ . My idea is to use the closed form $$T_n(x) = \frac{z_1^n + z_2^n}{2}, $$ ...