All Questions
Tagged with chebyshev-polynomials approximation
21
questions
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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
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65
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How is the Chebyshev polynomial approximation defined when using the spectral method to numerically solve PDEs?
I am studying spectral methods for numerical solutions for PDEs. Currently, I am on a chapter that explains how to use Chebyshev polynomials to solve non-periodic boundary value problems.
I understand ...
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75
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Coefficient of $x^n$ in Legendre series expansion
Suppose we are approximating a function $f$ with a Legendre series of order $N$, namely
$$
f(x) \approx \sum_{n=0}^N c_n P_n(x) \equiv f_N(x)
$$
where $P_n(x)$ is the $n^{th}$ Legendre polynomial and
$...
- 31
1
vote
1
answer
155
views
Show that for Chebyshev polynomials satisfy $(1-x^2)T'_k + kxT_k -kT_{k-1} = 0$
$T_n(x)$ is the Chebyshev polynomials of the first kind. I was wondering how to show the following recurrence relationship:
$$ (1-x^2)T'_k + kxT_k -kT_{k-1} = 0$$
This recurrence form also shows up as ...
- 553
4
votes
5
answers
437
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Efficiently approximating the integral of $\operatorname{erf}(x y) \exp(-x^2)$
I need to efficiently approximate this integral which represents the Gaussian-weighted area of a right triangle whose three points are the origin plus two points that form a vertical "edge", ...
- 163
2
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1
answer
48
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How to show the approximation of $1-T_{1 / L}(1 / \delta)^{-2}$ when $L$ is large and $\delta$ is small?
I met this approximation problem in eq(2) of this paper, stating that the approximation is $\left(\frac{\log (2 / \delta)}{L}\right)^{2}$.
As stated in the question, $T_L(x)$ is the $L_{th}$ Chebyshev ...
- 67
3
votes
0
answers
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
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2
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516
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Chebyshev approximation - from any equation
So let's say I have a function:
$f(x) = {e^x}$
How do I use Chebyshev polynomials up to order 4, to find the corresponding
coefficients? how do I make an approximation equation using these 4 ...
- 25
1
vote
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answers
61
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Chebyshev Approximation : concentrating nodes on specific intervals
I have a smooth function :
$$\frac{1}{2\pi} \sin(2\pi x )~,$$
that I would like to approximate on an interval $[-a, b]$. However I already know that all the inputs are guarantied to be close to an ...
- 11
1
vote
0
answers
477
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Chebyshev approximation for large interval
In the context of neural networks and cryptography, I would like to approximate some activation functions. However, I need to approximate them into polynomial forms for my purpose.
It seems that ...
- 133
1
vote
1
answer
406
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Minimize integral of polynomial interpolation error
Given $\mathrm{f}\in C\left[-1,1\right]$ solve:
$$
\min\int_{-1}^{1}\sqrt{\, 1 - x^{2}\,}\,\,\left\vert\,\mathrm{f}\left(x\right)-\mathrm{p}_n\left(x\right)\,\right\vert^{\,2}\,\mathrm{d}x
$$
where $\...
- 35
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votes
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295
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Find the cubic near-minimax or Chebyshev approximation for $f(x)=\sin(x)$
Find the cubic near-minimax or Chebyshev approximation for $f(x) = \sin(x)$ on the interval $[0,\frac{\pi}{2}]$.
Attempt: The first four Chebyshev polynomials are
\begin{align} T_0(x)&=1,\\ ...
- 10.8k
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votes
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71
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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 ...
- 1,011
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1
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66
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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 ...
