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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A question about extending polynomial span to monomial basis

I have a final next week and our instructor gave us some examples with solutions but I could not understand some operations. Inner product is $$(p,q)=\int_{-1}^{1} p(t)q(t)dt$$ $W = Span\{1,t,t^2\}$ ...
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Is this true for $n\geq 0$: $\lim\limits_{x\to \infty} e^{-x^2}\frac d {dx}\frac{T_n}{T_{n+1}}e^{x^2}=1 $?

Let $T_n$ denote Chebyshev polynomials of the first kind, I w'd like to show the below limit if it is true which it's hold for few first of them then how do i show this if it is true for $n\geq0$ ...
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Finding Points for Polynomial Interpolation

I'm studying for a final and came across the following question: Question In a discussion with my professor, he said using Chebyshev polynomials would be messy and unwieldy and encouraged another ...
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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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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 ...
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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 ...
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Generalization of identities for values of Chebyshev polynomials

Using the theory of Pell equations and the fact that the discriminant $5$ has both the forms $k^2 + 4$ and $k^2 - 4$, I stumbled across a proof of the identities $$ (-1)^n T_{2n} (\tfrac{i}{2}) = T_n ...
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Generalisation of Chebyshev minimax property

$\DeclareMathOperator*{\argmin}{arg\,min}$ The Chebyshev polynomials $$T_n(x) := \cosh(n \, \cosh^{-1}(x))$$ (with potentially complex $\cosh^{-1}(x)$) are well known to satisfy $$ \frac{T_n\left(\...
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Are there any alterations for the Chebyshev Differentiation Matrices on an arbitrary domain [a,b]?

I'm implementing the Chebyshev collocation method for solving PDEs, more specifically the shallow water equations. I know the Chebyshev differentiation matrices (or differential operators) are, for ...
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Integrate $I=\int_0^{\pi} \left(\cos(\theta)\right)^n \cos(p\theta) d\theta$

Is there any standard solution or way to solve the following integration $$I=\int_0^{\pi} \left(\cos(\theta)\right)^n \cos(p\theta) d\theta$$ where, $n=0, 1, 2,\dots$ and $p=0, 1, 2,\dots$ and $p>...
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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
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Reduced Chebyshev approximation?

Few days ago my teacher mentioned a method of approximation for a function. I think it was called "reduced chebyshev approximation", where you find the taylor series of degree $N$ then subtract from ...
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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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If a function is odd/even, then its best polynomial approximation is also odd/even. [duplicate]

If $f$ ∈ $C([a,b])$ is an even/odd function, then show that the best approximation among the polynomials of degree n is also even/odd. I'm almost certain that to show this directly, you should ...
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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}...
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Finding the dual of a LP

I am trying to find the dual of the following linear program. The primal LP's purpose is to find the lowest possible L1 (sum of absolute values of coefficients) of a degree $d$ polynomial such that (1)...
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Reverse economization of Chebyshev series

Suppose I have some function which is represented as converging series of Chebyshev polynomials of first kind in $[-1;1]$: $$ f(x)=\sum\limits_{n=1}^\infty a_n T_{2n}(x) $$ I need to transform this ...
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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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Divergence of squared sum of Chebyshev Polynomials $\equiv L+R$ has empty point spectrum

The Chebyshev Polynomials of the second kind $U_n$ are the solutions of the differential equation $$(1-x^2)U_n''(x)-3xU_n'(x)+n(n+2)U_n(x)=0$$ Alternatively they are defined inductively: $$U_0(x)=1 \...
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Trigonometric Function Simplification: $T_2 (x) = \cos (2 \arccos x)$

Let $T_n (x) = \cos (n \arccos x)$ where $x$ is a real number, $x \in [–1, 1]$ and $n$ is a positive integer. Show that $$T_2 (x) = 2x^2 – 1.$$ My attempt: $T_2 (x) = \cos (2 \arccos x)$ Because ...
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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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What happens to Chebyshev polynomials integration when n=1

The integration of Chebyshev polynomials of the first kind has the following value, $$\int T_{n}(x) \, dx = \frac{1}{2} \, \left( \frac{T_{n+1}(x)}{n+1} - \frac{T_{n-1}(x)}{n-1} \right)$$ what happens ...
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Chenge weight function in shifted orthogonality

We know that in Chebyshev orthogonal polynomial the weight function is $$\frac{1}{\sqrt{1-x^2}}$$ in interval $[-1,1]$. Do in shifted chebyshev orthogonal as example for interval $[0,1]$ the weight ...
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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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Different domains and ranges of the same Chebyshev polynomial?

Suppose I have a Chebyshev polynomial $T_2=\cos{(2\arccos(x))}$. Its range is $[-1,1]$ and its domain is $[-1,1]$. At the same time: $T_2=\cos{(2\arccos(x))}=2\cos^2{(\arccos(x))}-1=2x^2-1$ which has ...
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chebyshev nodes - formula for general interval [closed]

Could you help me write elegant form of formula for Chebysev nodes in $[a,b]$ ? Size of vector of points is $n$. I am working with octave, but the most important thing is: How to formulate it clear ...
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Roots of the Chebyshev polynomials of the second kind.

It is known that the roots of the Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$ and they are simple (of multiplicity one). I have noticed that the roots ...
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Derive recurrence relation for Chebyshev polynomials from generating function

Hej, I have a question about the following problem: Derive a recurrence formula for $m \ge 0$ given the generating function formula $$ \frac{1}{1-2xt+t^{2}}=\sum_{m=0}^\infty U_m(x)t^{m}. $$ What I ...
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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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How can I change the variable in this integral?

I have the following equality $\int_{0}^{\pi}cos(nt)cos(mt)dt=0$ (if $m\neq n$) This is in the context of Chebyshev polynomials and the book states the following to deduce ortoghonality. "the ...
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Expansion of cosine function by shifted chebyshev polynomials

$f(t)=f_1(t)+f_2(t) $ $w_1=1$, $w_2=6$ $f_1(t)=A \cos(w_1 t)$ $f_2(t)=B \cos(w_2 t)$ $A$ and $B$ are constant $2 \times 2$ matrix After normalizing the above equation by $t=T*y$ where ...
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prove two functions are orthogonal

I am new to this topic. Generally, what is the process of proving two polynomials orthogonal to each other on some given interval? Feeding some input and see that inner products are zero can probably ...
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relation between first kind Chebyshev poly and second kind Chebyshev poly

How do you prove following relation between Chebyshev poly of first kind and Chebyshev poly of second kind: $$dT_n(x)/dx=nU_{n-1}(x)$$
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Chebyshev Polynomials of 1/n

The Chebyshev polynomials of the first kind are defined by the recurrence relation: $$ T_n(x) = 2xT_{n-1}(x) - T_{n-2}(x)$$ By using these polynomials, you can multiply the frequency of a cosine by ...
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Building a cubic function with integer coefficients and trigonometric roots

I want to find the answer to the following problem: Construct a cubic polynomial with integer coefficients, whose roots - $\cos{\frac{2 \pi}{7}}$, $\cos{\frac{4 \pi}{7}}$ and $\cos{\frac{6 \pi}{7}}$. ...
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Definite Integals Obtained From Approximations Using Chebyshev Polynomial

Was trying to use chebyshev polynomial to obtain a cubic approximation to $f(x)=\frac{1}{x} $ I did it over the interval $[-1,1] $ Solving gave me the following four definite integrals: $C_0=\frac{...
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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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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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How to Change the Interval of Interpolation from [-1,1] to [a,b] for Chebyshev Nodes

(According to this website:http://fac-staff.seattleu.edu/difranco/web/Math_371_W11/Files/Chebyshevnodes.pdf) Between [-1,1], the Chebyshev Nodes are given as: $x_k = \cos\Big((2k-1)\pi/2n)\Big), k=...
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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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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 ...
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Chebyshev Interpolation

I'm studying Chebyshev Interpolation at the moment. Here n points $$x_1, x_2,..., x_n$$ are chosen within the interval $$-1 \le x \le 1$$ Using the Lagrange formula and minimising, we get: $$\prod_{...
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Differentiation using Chebyshev polynomials

I have task - find derivative (degree N) using Chebyshev polynomials. My issue is I can't find related theory. Give me a hint (book name, article, example), please. I'm very appreciative.
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A closed form for the coefficients of Chebyshev polynomials

The Chebyshev polynomials are defined recursively: $T_0(x)=1$; $T_1(x)=x$; $T_n(x)=2xT_{n-1}(x)-T_{n-2}(x)$ I have been trying to find a closed form for the coefficient on the monomial $x^j$ of the $...
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How to use Chebyshev Polynomials to approximate $\sin(x)$ and $\cos(x)$ within the interval $[−π,π]$?

I have approximated $\sin(x)$ and $\cos (x)$ using the Taylor Series (Maclaurin Series) with the following results: $$f(x)=f(0)+\frac{f^{(1)}(0)}{1!}(x-0)+\frac{f^{(2)}(0)}{2!}(x-0)^2+\frac{f^{(3)}(0)...
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Inverse Chebyshev Recurrence

The Chebyshev polynomials (of the first kind) are a sequence of polynomials defined recursively by $$ \begin{cases} T_{0}(x) = 1 \\ T_{1}(x) = x \\ T_{n}(x) = 2xT_{n-1}(x) - T_{n-2}(x) \end{cases} $$ ...
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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 ...
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Transform complex trigonometric expression with $\arccos$

In the proof that the poles of a Chebyshev filter lie on an ellipse, there is the following transformation, for the $s$ values correspondant to the poles. From (1) $$s_{pm} = j \cos \left[ \...
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How is optimal coordinates change chosen for Chebyshev expansion?

I'm looking into SLATEC implementation of Bessel function $J_0$ computation (readable in C in GSL), namely at its part for arguments in interval $(0,4)$. There a Chebyshev expansion is used, but the ...
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A cheap error estimate and a costly doubt

Carl de Boor poses the following problem in his A Practical Guide to Splines (1978 - Chapter II, p. 38, problem 4): The calculation of $||g|| = \max\{|g(x)| : a \le x \le b\}$ is a nontrivial task. ...