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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29 views

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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Series Representation of the Glasser function: $\text G(x)\mathop=\limits^\text{def} \int_0^x \sin(t\sin(t))dt\sim2\sqrt{\frac x\pi}$

Here is an uncommon special function called the Glasser function as referenced by Wolfram Mathworld. which is defined as: $$\text G(x)\mathop=^\text{def} \int_0^x \sin(t\sin(t))dt\sim2\sqrt{\frac x\pi}...
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Generating function for Legendre functions of second kind with half integer order

Good morning, I am looking for a generating function $f(t,z)$ for the Legendre functions of second kind $Q_{n-\frac{1}{2}}(z)$. $$f(t,z) = \sum_{n=0}^{\infty} t^n Q_{n-\frac{1}{2}}(z) $$ Can anyone ...
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On the parity of the first kind Chebyshev polynomial over the positive integers

Let us consider the first kind Chebyshev polynomial over the positive integers $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$ with $n>2$ is an odd number. We have $$T_0(x) = 1$$ $$T_1(x) = x $$ $$T_3(x) = 4x^...
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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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Coefficients of series solutions of Chebyshev Differential Equation

Let $\alpha>0$ and $\alpha\notin\mathbb{N}$. If $\sum_{k=1}^\infty a_kx^k$ is the power series solution satisfying the Chebyshev Differential Equation, then $$a_{k+2}=\frac{(k-\alpha)(k+\alpha)}{(k+...
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Maximum of sum harmonic sines under phase shift

Given the sum of $N$ harmonic sinusoids with weights $a_n > 0$: $$ x(\theta) = \sum_{n = 1}^N a_n \sin(n \theta + \phi_n) $$ How to determine the absolute maximum of $x$, i.e. $\max_\theta \lvert ...
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Integral of Legendre and Chebyshev polynomials.

I am trying to expand Legendre polynomials into Chebyshev polynomials, shown as: $$P_{n}(x)=\sum_{k=0}^{n}a_{k}T_{k}(x), $$ where $P_{n}$ is Legendre polynomials and $T_{k}$ is Chebyshev polynomials, ...
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How we generalize the cartesian form of epicycloids?

I have the following parametric form of epicycloids: $x(t)=\frac{a\cdot\cos t+\cos(a\cdot t)}{1+a}$ $y(t)=\frac{a\cdot\sin t+\sin(a\cdot t)}{1+a}$ where $a=2,3,4,\ldots$ is a variable that ...
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Relationship between Chebyshev polynomials and square roots ($\sqrt{3}+\sqrt{2}=\frac{1}{\sqrt{T_1(5)-\sqrt{T_1(5)^2-1}}}$ etc.)

(If my English is strange, I would appreciate it if you could correct it.) There seems to be a property about the sum of square roots. (This is almost self-explanatory.) let $ a,\ b,\ t \in \mathbb{N}^...
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Is it possible to approximate multivariate functions using Chebyshev polynomials?

I have a 2D unstructured mesh with pressure defined at each point. I want to compute the local gradient of pressure (using 10-20 neighbouring points) in 2D space using better-conditioned polynomials ...
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Do $\{T_{n}\left(\frac{x^2}{2}-1\right)\}_{n=1}^\infty$ form a basis for the even degree polynomials?

As we know, Chebyshev polynomials form a complete set of independent functions, i.e. they form a basis for the set of polynomials. Let us consider a class of shifted Chebyshev polynomials of the first ...
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Quadrature for integral of exponential of Chebyshev polynomial?

For Chebyshev polynomials: \begin{align*} T_0(x) &= 1 \\ T_1(x) &= x \\ T_2(x) &= 2x^{2} - 1 \\ & \,\,\,\vdots \end{align*} is there a good way to approximate the following integral?: \...
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Energy integral using Chebyshev coefficients

I am trying to evaluate the following integral \begin{equation} E = \int_{-1}^1 f(x)g(x) \ dx \end{equation} I have the values for Chebyshev coefficients of the function $f(x) = \sum_{m=0}^{N-1} \hat{...
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Chebyschev's Polynomials and its Application

Consider approximating the function $f(x)=x^{3}-x$ by a polynomial $P_{2}(x)=a_{2} x^{2}+$ $a_{1} x+a_{0}$ which minimizes $$ E_{2}\left(a_{0}, a_{1}, a_{2}\right)=\int_{0}^{1}\left[f(x)-P_{2}(x)\...
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How to prove by induction all Chebyshev polynomials of the first kind $T_n$ when $n\geq 1$ have a positive leading coefficient??

Use the first kind Chebyshev polynomial $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$ to show how the leading coefficient is always positive $(1, 2, 4, 8, 16, 32...)$ when $n\geq 1$ using proof by induction $$...
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An integral with Chebyshev polynomials

I am posting this question on behalf of a friend of mine, who studies mathematics and isn't proficient in English. In his project work, he encountered the following integral: $$ \int_{-1}^1 x^k U_n(x) ...
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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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42 views

Orthogonality of Chebyshev Second Kind

I was reading on Chebyshev functions, and I found lots of resources on proving the orthogonality of Chebyshev polynomials of the first kind: $\int_{-1}^1 T_m(x) T_n(x) \frac{dx}{\sqrt{1-x^2}} = \...
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Matlab Problem: Badly scaled matrix, very small condition number RCOND when using Chebyshev discretization

I am using MATLAB for the following problem. I have the following problem statement: LHS * q = RHS * f. This can be rewritten to q = H *f with H = LHS\RHS; Hereby q and f are vectors, LHS and RHS ...
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Chebyshev’s inequality meteorite question

Enceladus is the sixth-largest moon of Saturn. Enceladus is often called the protector of Saturn. Many meteorites that were supposed to fly to Saturn will be affected by the Enceladus's gravity and ...
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Proving that the sum of $\cos k \theta_\mu \cos\theta_\nu$ from k=0(this term is to be halved) to n-1 equals to $\frac{1}{2}n\delta_{\nu\mu}$.

Been trying a while to prove that \begin{equation*}\sideset{}{'} \sum_{k=0}^{n-1} \cos k\theta_\nu\hspace{0.05cm} \cos k\theta_\mu=\frac{1}{2}n\delta_{\nu\mu}, \hspace{0.3cm}\nu,\mu=1,2,\dots,n, \end{...
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Do shifted Chebyshev polynomials form a complete set of independent functions?

Do Chebyshev polynomials form a complete set of independent functions? If yes, what can we say about their shifted versions? E.g. shifted Chebyshev polynomials of the first kind are defined as $$T_{n}^...
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Are the coefficients of Chebyshev expansion decreasing with their index $n$?

Consider the expansion of $f \in [-1,1]$ in Chebyshev polynomials of the first kind $T_n(x)$: $$f(x)=\sum_{n=0}^\infty a_{n} T_n(x), \quad a_n=\frac{2}{\pi} \int_{-1}^1 \frac{f(x) T_n(x)}{\sqrt{1-x^2}}...
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Recostruct a function from the coefficients of its Chebyshev expansion

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 know ...
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Why $w_i=\pi/n$ Chebyshev–Gauss quadrature?

In Chebyshev–Gauss quadrature we have $$\int _{-1}^{+1}{\frac {f(x)}{\sqrt {1-x^{2}}}}\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})$$ where $$x_{i}=\cos \left({\frac {2i-1}{2n}}\pi \right)$$ and the weight ...
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Increase resolution of Chebyshev discretization locally

I am currently discretizing a PDE using the Chebishev discretization in Matlab as seen below. Currently I am increasing the resolution of the discretization in the whole grid by the choice of N, the ...
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Knowing $\cos\theta$, can $\cos(n\theta)=\cos(\pi k)$?

$\cos(\pi k)=1$ or $-1$. After expressing $\cos(n\theta)$ in terms of $\cos\theta$, I have found that $\cos(n\theta)=\sum^{\lfloor{\frac{n}{2}}{\rfloor}}_{l=0}{n\choose2l}(-\frac{8}{9})^{l}(-\frac{1}{...
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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 ...
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20 views

Recurrent formula for Chebyshev polynomial on general range

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} ...
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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 ...
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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 $x^3$...
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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|
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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{...
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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_{...
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Rescaled Chebyshev Polynomials have the smallest maximum

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 $\arg \min_{p \in P^n}( \max_{x\in[m, M]}p(x))$. ...

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