Faber polynomials

In mathematics, the Faber polynomials P_m of a Laurent series

${\displaystyle \displaystyle f(z)=z^{-1}+a_{0}+a_{1}z+\cdots }$

are the polynomials such that

${\displaystyle \displaystyle P_{m}(f)-z^{-m}}$

vanishes at z=0. They were introduced by Faber (1903, 1919) and studied by Grunsky (1939) and Schur (1945).

References

• Curtiss, J. H. (1971), "Faber Polynomials and the Faber Series", The American Mathematical Monthly, 78 (6), Mathematical Association of America: 577–596, doi:10.2307/2316567, ISSN 0002-9890, JSTOR 2316567
• Faber, Georg (1903), "Über polynomische Entwickelungen" (PDF), Mathematische Annalen, 57, Springer Berlin / Heidelberg: 389–408, doi:10.1007/BF01444293, ISSN 0025-5831
• Faber, G. (1919), "Über Tschebyscheffsche Polynome.", Journal für die reine und angewandte Mathematik (in German), 150: 79–106, ISSN 0075-4102, JFM 47.0315.01
• Grunsky, Helmut (1939), "Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen", Mathematische Zeitschrift, 45 (1): 29–61, doi:10.1007/BF01580272, ISSN 0025-5874
• Schur, Issai (1945), "On Faber polynomials", American Journal of Mathematics, 67: 33–41, doi:10.2307/2371913, ISSN 0002-9327, JSTOR 2371913, MR 0011740
• Suetin, P. K. (1998) [1984], Series of Faber polynomials, Analytical Methods and Special Functions, vol. 1, New York: Gordon and Breach Science Publishers, ISBN 978-90-5699-058-9, MR 1676281
• Suetin, P. K. (2001) [1994], "Faber polynomials", Encyclopedia of Mathematics, EMS Press