Chebyshev Nodes for Interpolation

[gucci1]

Hello all,

I'm not sure if this is the right sub-forum for this question, so please move my post if there is a better place for this.

I have been studying interpolation of functions, and in particular, using Chebyshev nodes rather than equidistant points in the creation of an approximation polynomial. In my textbook, it states that "it is not difficult to construct examples where Chebyshev interpolation would not do well." However, I am having considerable difficulty coming up with an example on my own.

I understand that in the Runge function, the reason that Chebyshev nodes do a better job than equidistant points is that $\frac{f^{(n+1)}}{(n+1)!}$ grows near the endpoints of the interval [-1,1], and this happens to be where the Chebyshev nodes are more concentrated. But I don't know how to create an example where this is not the case.

I hope that this question makes sense, and I will be happy to explain further if something is unclear. Thank you in advance for any help you can offer,

gucci

 

[jvicens]

_girl

Dear gucci_girl,

Thank you for your question. I can understand your confusion about finding an example where Chebyshev interpolation does not perform well. This is because Chebyshev interpolation is known to be a highly accurate method for approximating functions, especially compared to equidistant points.

However, one example where Chebyshev interpolation may not perform as well is when the function being approximated has a singularity or a sharp change in behavior near the endpoints of the interval. In this case, the Chebyshev nodes may not be as concentrated near the problematic points, leading to a less accurate approximation compared to equidistant points.

Another situation where Chebyshev interpolation may not be the best choice is when the function is not smooth or has discontinuities. In these cases, the Chebyshev nodes may not be able to capture the behavior of the function accurately, resulting in a less accurate approximation compared to equidistant points.

I hope this helps in your understanding of Chebyshev interpolation and its limitations. If you have any further questions, please feel free to ask. Best of luck with your studies.