Update README.md

README.md

@@ -12,16 +12,19 @@ Install the package `python -m pip install rexpi` and run numerical examples fro
 
 We consider rational approximations $$r(\mathrm{i} x) \approx \mathrm{e}^{\mathrm{i}\omega x},\quad\quad\textrm{for}\quad x\in[-1,1]\quad\textrm{and a frequency}\quad\omega>0,$$ where $r$ is a unitary $(n,n)$-rational function, i.e., $r=p/q$ where $p$ and $q$ are polynomials of degree $\leq n$ and $|r(\mathrm{i} x)|=1$ for $x\in\mathbb{R}$. Our focus is best approximations in a Chebyshev sense which minimize the uniform error $$\max_{x\in[-1,1]}| r(\mathrm{i} x) - \mathrm{e}^{\mathrm{i}\omega x} |.$$
 
-### Best approximations
+### Best approximations and equioscillating phase errors
 
-Following a recent work[^JSxx], the unitary best approximation to $\mathrm{e}^{\mathrm{i}\omega x}$ is uniquely characterized by an equioscillating phase error. In particular, unitary rational functions are of the form $r(\mathrm{i} x) = \mathrm{e}^{\mathrm{i}g(x)}$ for $x\in\mathbb{R}$ where $g$ is a phase function, and we introduce the phase error $g(x) - \omega x$ in addition to the approximation error $r(\mathrm{i} x) - \mathrm{e}^{\mathrm{i}\omega x}$. The phase and approximation error satisfy the relation $|r(\mathrm{i} x) - \mathrm{e}^{\mathrm{i}\omega x}| = 2|\sin((g(x) - \omega x)/2)|$, and for the case that the approximation error is non-maximal, i.e., $\max_{x\in[-1,1]}| r(\mathrm{i} x) - \mathrm{e}^{\mathrm{i}\omega x} |<2$, the best unitary rational approximation to $\mathrm{e}^{\mathrm{i}\omega x}$ corresponds to the phase function $g$ which minimizes the phase error $\max_{x\in[-1,1]} |g(x) - \omega x|$. Following[^JSxx], the corresponding phase error $g(x)-\omega x$ equioscillates at $2n+2$ nodes $\eta_1< \eta_2< \ldots <\eta_{2n+2}$, namely, $$g(\eta_j) - \omega \eta_j = (-1)^{j+1} \max_{x\in[-1,1]}| g(x) - \omega x |,\quad\quad j=1,\ldots,2n+2.$$ In the present package we introduce numerical algorithms to compute unitary best approximations mostly based on consequences of this equioscillation property, and certainly relying on the result that the unitary best approximation is unique and non-degenerate.
+Following a recent work[^JSxx], for $\omega\in(0,\pi(n+1))$ the unitary best approximation to $\mathrm{e}^{\mathrm{i}\omega x}$ is uniquely characterized by an equioscillating phase error. In particular, unitary rational functions are of the form $r(\mathrm{i} x) = \mathrm{e}^{\mathrm{i}g(x)}$ for $x\in\mathbb{R}$ where $g$ is a phase function, and $g(x) - \omega x$ is the phase error of $r(\mathrm{i} x) \approx \mathrm{e}^{\mathrm{i}\omega x}$. 
+The phase error equioscillates at $2n+2$ nodes $\eta_1< \eta_2< \ldots <\eta_{2n+2}$ if $$g(\eta_j) - \omega \eta_j = (-1)^{j+1} \max_{x\in[-1,1]}| g(x) - \omega x |,\quad\quad j=1,\ldots,2n+2.$$
+E.g. the phase and approximation error of the unitary rational best approximation for $n=4$ and $\omega=2.65$ generate by our `brib` algorithm.
+![errors](https://github.com/newbisi/rexpi/blob/main/docs/errors.png)
 
 As a consequence of the equioscillation property, the unitary best approximation attains $2n+1$ points of interpolation $x_1 <\ldots < x_{2n+1}$ in the sense of $r(\mathrm{i} x_j) = \mathrm{e}^{\mathrm{i}\omega x_j}$. The existence of such interpolation nodes motivates the `brib` algorithm introduced in the present package - a modified BRASIL algorithm which aims to compute a uniform best approximation by rational interpolation and by iteratively adapting the underlying interpolation nodes. We remark that an earlier algorithm[^Fra76] and the 'Second Direct Method' in[^Ma63] are based on similar ideas, but suggest different approaches to adapt interpolation nodes which will be tested in future versions.
 In addition, the present package also provides a variant of the AAA-Lawson method to compute approximations to $\mathrm{e}^{\mathrm{i}\omega x}$. It has been shown recently in[^JS23] that rational interpolants and rational approximations constructed by the AAA-Lawson algorithm are unitary, and thus, these approaches fit well to compute unitary rational best approximations.
 
 #### Numerical illustration of the equioscillating phase error and the approximation error
-To generate these plots we used the `brib` algorithm with $n=4$ and $\omega=2.65$.
-![errors](https://github.com/newbisi/rexpi/blob/main/docs/errors.png)
+
+
 
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