All Questions
Tagged with chebyshev-polynomials approximation-theory
24
questions
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Best uniform approximation of complex exponential function $e^z$ over unit disc in complex plane
It is known that the best uniform approximation for a real function defined in interval $[1,-1]$ is via the Chebyshev polynomials. ([see optimal polynomials])1. Such polynomials are also called min-...
- 125
2
votes
1
answer
52
views
Compute the correction of a Chebyshev approximation using the Clenshaw summation formula
Assume you have a Chebyshev approximation of a function $f(x)$ evaluated using the Clenshaw summation method, up to polynomial order $N$:
$$
f(x) = \sum_{k=0}^{N-1} a_k T_k(x) = (a_0 - y_2)T_0(x) + ...
- 21
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43
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Bounds of polynomial approximation of a function of many variables using Jackson inequality
There is an approximation of a multivariate function by a Chebyshev polynomial of degree n. One needs to understand how the approximation error behaves depending on the degree of the polynomial or ...
- 1
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48
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Chebyshev approximation for bivariate function
I read the paper.
I am a litte bit confused regarding formulation of Chebyshev approximation for bivariate function(See photo). There is only one integral over variable x. Should it be in formula one ...
- 1
0
votes
1
answer
106
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Convergence rate for Chebyshev polynomials to approximate $\text{erf}(x)$ on a subset of $\mathbb{R}$
Let $[-\alpha, \alpha] \subset \mathbb{R}$, and let
\begin{equation}
\text{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}dt.
\end{equation}
given projections of $\text{erf}(x)$ onto the first $k$ ...
- 533
6
votes
2
answers
251
views
Polynomial that grows faster than any other polynomial outside $[−1,1]^n$
Consider this statement: "Chebyshev polynomials increase in magnitude more quickly outside the range $[−1,1]$ than any other polynomial that is restricted to have magnitude no greater than one ...
- 543
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158
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Question on convergence of Chebyshev series
I have written a script that plots some function and its truncated Chebyshev expansion in $[-1,1]$, which is given by
$$
\sum a_nT_n(x) \quad \text{with} \quad a_n = \int_{-1}^{1}T_n(x)f(x)/\sqrt{1-x^...
- 499
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202
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Bounds for Chebyshev polynomials outside [-1,1]
It is well known that, for all $x\in [-1,1]$ and for all $j>0$, the Chebyshev polynomials of the first and second kind satisfy
$$
|T_j(x)|\leq 1
,\qquad
|U_j(x)|\leq j+1.
$$
I am wondering about ...
- 1,420
3
votes
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85
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Chebyshev coefficients of Chebyshev interpolants
I am wondering whether there is a known closed form for the $k$th Chebyshev coefficient of a $n$th Chebyshev interpolant, that is,
$$\int_{-1}^1T_k(x)L_{n,i}(x)\frac{dx}{\sqrt{1-x^2}}$$
where $T_k$ is ...
- 1,147
1
vote
0
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70
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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
$$
E_{2}\left(a_{0}, a_{1}, a_{2}\right)=\int_{0}^{1}\left[f(x)-P_{2}(x)\...
- 1,023
4
votes
2
answers
1k
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The relationship between the best uniform approximation and chebyshev interpolant
I am learning Approximation Theory. I know one polynormial of degree at most $n$ is the best uniform approximation of the function $f \in \mathcal C[a,b]$ if and only if there is exist a set of $n+2$ ...
- 705
3
votes
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257
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Converting Chebyshev expansion into a regular polynomial
So I have a very very basic technique for converting shifted Chebyshev polynomials to regular polynomials but it seems to have a large issue with numerical stability and I don't completely understand ...
- 461
3
votes
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452
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Error estimate of a Chebyshev polynomial approximation.
I am trying to approximate a function $f(x)$ on $[-1, 1]$ using Chebyshev's polynomial of the first kind.
$$ f(x) \approx \sum_{i=0}^N a_iT_i(x) $$
What is the error of this approximation? Is it the ...
- 310
1
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498
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How to Find the Coefficients For the Chebyshev Expansion of a Function
Find and building a Fourier series of a function $f(x)$ on an arbitrary interval $[a,b]$ is explained here. I know that for Chebyshev series, the expansion is $$f(x) \sim \sum_{i=0}^{N} c_i T_i(x)$$ ...
- 788
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votes
1
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37
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Show that the sign of the sequence is alternating [closed]
I'm solving the following (numerical analysis) problem.
Let $f(x)$ be a polynomial of degree $n$ such that there exist exactly $n+1$ distinct points $x_0,x_1\cdots,x_n$ in the interval $[-1,1]$ ...
- 451