[math-fun] Complex Cheby (Chufty?!?)
Cheby expansion: When complex z, |z|=1, T_k(x) = realpart(z^k) = (z^k+z^(-k))/2 (i.e., z=x+i*y) By *starting* with approximations in the complex domain, Trefethen shows how real Cheby approximations fall out in the wash, instead of appearing on stone tablets handed down from on high. Thus, the choice of x's as x-projections of *evenly spaced* points around the unit circle no longer seems forced -- indeed, it's *obvious*. A treatment of Cheby's in the *complex* domain also makes obvious the use of FFT's in various calculations involved with Cheby's. Finally, Trefethen shows in other papers that simply choosing these x-projections of equally spaced points on the unit circle *automatically* achieves the benefits of Cheby's when used for Lagrange interpolation, Newton divided differences, etc. So the problems with Lagrange, etc., aren't Lagrange's or Vandermonde's fault, but the fault of the unfortunate choice of equally spaced x-coordinates. I realize that Chebyshev polynomial approximations may not have originally been derived this way, but why do we still have to keep teaching them as if the complex domain never existed? For example, neither https://en.m.wikipedia.org/wiki/Chebyshev_polynomials nor https://en.m.wikipedia.org/wiki/Approximation_theory even hints at the elegant treatment that a view from the complex plane enables. http://people.maths.ox.ac.uk/trefethen/publication/PDF/1983_8.pdf Chebyshev Approximation on the Unit Disk Lloyd N. Trefethen
Since you mentioned Lloyd Trefethen and Chebyshev polynomials, if you haven't seen it, you should look at his excellent package Chebfun: http://www.chebfun.org/ for doing really well-principled numerical calculations. The main package is for matlab, but there are implementations as C libraries and for python, for example. Victor On Mon, Jan 28, 2019 at 12:50 PM Henry Baker <hbaker1@pipeline.com> wrote:
Cheby expansion:
When complex z, |z|=1,
T_k(x) = realpart(z^k) = (z^k+z^(-k))/2
(i.e., z=x+i*y)
By *starting* with approximations in the complex domain, Trefethen shows how real Cheby approximations fall out in the wash, instead of appearing on stone tablets handed down from on high. Thus, the choice of x's as x-projections of *evenly spaced* points around the unit circle no longer seems forced -- indeed, it's *obvious*.
A treatment of Cheby's in the *complex* domain also makes obvious the use of FFT's in various calculations involved with Cheby's.
Finally, Trefethen shows in other papers that simply choosing these x-projections of equally spaced points on the unit circle *automatically* achieves the benefits of Cheby's when used for Lagrange interpolation, Newton divided differences, etc. So the problems with Lagrange, etc., aren't Lagrange's or Vandermonde's fault, but the fault of the unfortunate choice of equally spaced x-coordinates.
I realize that Chebyshev polynomial approximations may not have originally been derived this way, but why do we still have to keep teaching them as if the complex domain never existed?
For example, neither
https://en.m.wikipedia.org/wiki/Chebyshev_polynomials
nor
https://en.m.wikipedia.org/wiki/Approximation_theory
even hints at the elegant treatment that a view from the complex plane enables.
http://people.maths.ox.ac.uk/trefethen/publication/PDF/1983_8.pdf
Chebyshev Approximation on the Unit Disk
Lloyd N. Trefethen
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