# Questions tagged [chebyshev-polynomials]

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are two sequences of orthogonal polynomials which are related to de Moivre's formula. These polynomials are also known for their elegant Trigonometric properties, and can also be defined recursively. They are very helpful in Trigonometry, Complex Analysis, and other branches of Algebra.

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### Any Python libraries for finding the coefficients of a polynomial in the Chebyshev basis? [closed]

I'm writing a program where I need to find the coefficients of an arbitrary polynomial in the Chebyshev basis, does anyone know of a library or script that does that so I won't have to implement it ...
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### Prove that there is exactly one element for interpolation in a Chebyshev system

Consider a Chebyshev system $g_0,...,g_n \in C[a,b]$ and $(n+1)$ value pairs where $x_i\neq x_j$ for $i\neq j$ that are all in $[a,b]$. Prove that there is exactly one element $g \in span(g_0,...,g_n)$...
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### Graph Neural Networks Spectral Methods: How is Chebyshev filter approximation formula derived?

I was reading the following paper: here. Context: For reference, the previous equation is: $$g_{\theta}(\Lambda) = \sum_{k = 0}^{K - 1} \theta_k \Lambda ^k$$ where the parameter $\theta \in R^K$ is ...
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### Are decompositions of products of Chebyshev polynomials (evaluated at a given point) into summands unique over extended integer rings?

The $i$-th Chebyshev polynomial of the 2nd kind is defined to be the polynomial $U_i(x)$ that satisfies \begin{equation} U_i(\cos \theta) \sin\theta = \sin(i+1)\theta. \end{equation} There exists an ...
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### The diophantine equation $T_{n}(x)=a$

Let us consider the Chebychev polynomial function $T_{n}(x)$ where $x$ is a positive integer variable and consider the diophantine equation $$T_{n}(x)=a......(*)$$ where $a$ is a positive integer. I ...
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### The relationship between the best uniform approximation and chebyshev interpolant

I am learning Approximation Theory. I know one polynormial of degree at most $n$ is the best uniform approximation of the function $f \in \mathcal C[a,b]$ if and only if there is exist a set of $n+2$ ...
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### I am asking if this quantity has a name or it is just a real sequence.

Let us consider the Chebychev polynomial function $T_{n}(x)$ where $n$ is a fixed positive integer called the degree and $x$ is the real variable. Let us consider $x$ as a positive integer variable ...
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### Finding the minimal number of nodes required to integrate $e^{-rt}$ via Chebyshev nodes.

I have a function $e^{-rt}$, defined on some interval $[a,b]$, and I want to integrate this function via Chebyshev nodes to some precision $\delta$. How would I go about finding the number of nodes ...
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### Converting Chebyshev expansion into a regular polynomial

So I have a very very basic technique for converting shifted Chebyshev polynomials to regular polynomials but it seems to have a large issue with numerical stability and I don't completely understand ...
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### Algorithm for Chebyshev series derivative

I have to evaluate the second derivative of a truncated Chebyshev series $$\frac{\mathrm{d}^2p_N}{\mathrm{d}x^2}=\frac{\mathrm{d}^2}{\mathrm{d}x^2}\left(\sum_{k=0}^N a_kT_k(x)\right)$$ in a given ...
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### Chebyshev Coefficients

I want to show that aj in this photo, but I am unsure of how to get there I know it is a minimum so it takes differentiation to get there but I am unsure how, any help would be appreciated, thanks ...
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### Damped Chebyshev interpolant

Suppose I construct an interpolating polynomial $p$ for $f$ at the degree $k$ Chebyshev nodes of the first kind. Then, even if $f(x) > 0$ for all $x$, we do not have that $p(x)\geq 0$. Represent $p$...
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### How to solve the recurrence relation of Chebyshev polynomials

The Chebyshev polynomials of the first kind are obtained from the recurrence relation \begin{aligned}T_{0}(x)&=1\\T_{1}(x)&=x\\T_{n+1}(x)&=2x\,T_{n}(x)-T_{n-1}(x)~.\end{aligned} Is it ...
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### Chebyshev polynomial factorization

I have a Chebyshev Polynomial $C_n$. I have to argue that $C_{n+1}$ can be written as: $$C_{n+1}(t) = 2^n (t - \theta_{0,0})(t - \theta_{0,1})\dots(t - \theta_{0,n}),$$ where the parentheses can be ...
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### Error estimate of a Chebyshev polynomial approximation.

I am trying to approximate a function $f(x)$ on $[-1, 1]$ using Chebyshev's polynomial of the first kind. $$f(x) \approx \sum_{i=0}^N a_iT_i(x)$$ What is the error of this approximation? Is it the ...
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### Divisibility of Chebyshev Polynomials

I was trying to solve a problem involving an Insect crawling on the Cartesian/Coordinate Plane. We have an insect on the origin of the coordinate plane, who remembers a particular angle $\theta.$ We ...
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### Prove the orthogonality relation of Chebyshev polynomials of the first kind

The Chebyshev polynomials of the first kind are obtained from the recurrence relation \begin{aligned}T_{0}(x)&=1\\T_{1}(x)&=x\\T_{n+1}(x)&=2x\,T_{n}(x)-T_{n-1}(x)~.\end{aligned} Prove ...
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### Chebyshev expansion of $f(x)=\frac{1}{1+(x-s)^2}$

The Chebyshev polynomials of the first kind are obtained from the recurrence relation \begin{aligned}T_{0}(x)&=1\\T_{1}(x)&=x\\T_{n+1}(x)&=2x\,T_{n}(x)-T_{n-1}(x)~.\end{aligned} I ...
I'm trying to compute Chebyshev polynomial coefficients on [a, b] and tried to change default recurrent formula $$T_{n+1} = 2xT_n - T_{n-1}$$ to $$T_{n+1} = (2x-(b+a))T_n - \frac{(b-a)}{4}^{2} T_{n-1} ... 1answer 316 views ### Monte-Carlo integration Let a function f to be x\in \left[a,b\right],\:0\le f\left(x\right)\le c. We want to calculate the approximation of the definite integral of the function in the range [a,b], we can suppose that ... 0answers 62 views ### Powers of x as Chebyshev polynomials I want to convert the first five terms of the Taylor series expansion for e^x into Chebyshev Polynomial, but it requires that I express the power of x as Chebyshev first, I've gotten it up to x^3... 0answers 45 views ### Minmax approximation using Chebyshev polynomial Find minmax approximation to f(x)=|x| in P_3 on [-1,1] .What is the minmax error I do not know how to do that ,i got hint that i have to use Chebyshev polynomial for approximating |x| 0answers 27 views ### Bound for coefficients of chebyshev interpolation in standard basis. Assume we have for some (sufficiently) smooth function h:[-1,1] \rightarrow \mathbb{C} the Chebyshev interpolation of degree n \in \mathbb{N} \begin{equation} P_n(x) = \sum_{i=0}^n c_iT_i(x). \end{... 1answer 33 views ### Can we define the generating function for all x and all t The Chebyshev polynomials of second kind are defined for any x \in \Bbb R (or even x \in \Bbb C), e.g. via the recurrence relation$$ U_0(x) = 1 \\ U_1(x) = 2x \\ U_{n+1}(x) = 2x U_n(x) - U_{...
Let $P^n$ be the set of polynomials $P^n \equiv \{p\, | \deg p = n, \, p(0) = 1\}$. Let $[m, M]$ be an interval where $0 \lt m \lt M$. I want to find $\arg \min_{p \in P^n}( \max_{x\in[m, M]}p(x))$. ...