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Approximation theory, which aims to provide the optimal polynomial function approximating the target function in a given domain such as $x\in[-1,1]$, has been well-developed for real variables. In particular, the Chebyshev polynomial plays an important role in that it often provides near-optimal performance.

My questions are: I) Are there studies/theories for approximation theory for complex variables (i.e., holomorphic functions, especially for $|x|<1$) and maybe some reference on this; II) What is the counterpart of Chebyshev polynomials in this case?

PS: I have also noticed other questions in the channel. Yet they do not address my problems directly.

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    $\begingroup$ Do you want to approximate holomorphic functions? Are your functions defined in a particular domain, maybe a disk? $\endgroup$
    – Ben McKay
    Commented Jan 26 at 10:13
  • $\begingroup$ Yes, that's what I meant. To be more specific, the requirements are variables inside a disk such as $|x|<1$, and here it is only interested in polynomial but not rational polynomial functions. $\endgroup$
    – ironmanaudi
    Commented Jan 26 at 10:17
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    $\begingroup$ The relationship between approximation theory and complex variables is so intimate that there is even an important branch of analysis centred on it--constructive function theory. The classic text is a three volume monograph under that name by I. P. Natanson. $\endgroup$
    – terceira
    Commented Jan 26 at 17:20

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The theory of approximation in the complex plane is almost as rich as the theory on the real interval. Some of the good books are:

D. Gaier, Lectures on complex approximation. Translated from the German by Renate McLaughlin. Birkhäuser Boston, Inc., Boston, MA, 1987.

J. L. Walsh, Interpolation and approximation by rational functions in the complex domain. Fourth edition. American Mathematical Society Colloquium Publications, Vol. XX. American Mathematical Society, Providence, RI, 1965.

In particular, Chebyshev's theorem on the existence and uniqueness of a polynomial of the best uniform approximation has been extended by Kolmogorov to arbitrary set in the complex plane. (Such a polynomial of degree $d$ exists and is unique when the set contains at least $d+2$ points. It is characterized by the property that there are at least $d+2$ points where $\max|f-P_d|$ is attained.)

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  • $\begingroup$ Thanks for the excellent answer! Can you provide the name of the best approximation polynomial function? I came across this article saying that the Faber polynomial provides the near-best uniform approximation. $\endgroup$
    – ironmanaudi
    Commented Jan 27 at 13:59
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    $\begingroup$ For the unit disk, both Faber polynomial, and Chebyshev polynomial is $z^d$. $\endgroup$
    – Alexandre Eremenko
    Commented Jan 27 at 15:24
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    $\begingroup$ For Faber, this immediately follows from definition. For Chebyshev, it follows from Kolmogorov's theorem which says that the Chebyshev polynomial of given degree d is unique, and is characterized by the property that the deviation achieves its maximum at at least d+2 points. $\endgroup$
    – Alexandre Eremenko
    Commented Apr 10 at 13:34
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    $\begingroup$ @108_mk: To my embarrassment, I do not know any reference on Kolmogorov, except his original paper published in Russian: Uspekhi Mat. Nauk, 1948, vol. 3, 1, 216-221. $\endgroup$
    – Alexandre Eremenko
    Commented Apr 10 at 13:49
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    $\begingroup$ Here is the Russian text: mathnet.ru/links/dfa99a668b3274b5931db279e1e8455f/rm8691.pdf $\endgroup$
    – Alexandre Eremenko
    Commented Apr 10 at 13:52
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For general complex valued functions on a disk, a good alternative would be the Zernike polynomials.

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One place to start looking might be Trefethen's work, in particular Chebfun https://www.chebfun.org and the rational approximation AAA, as discussed in this retrospective: https://people.maths.ox.ac.uk/trefethen/nak_sete_tref_revised.pdf

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  • $\begingroup$ These approximants are not optimal, though, I believe. $\endgroup$
    – Federico Poloni
    Commented Apr 10 at 22:45

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