Behavior of polynomials of best uniform approximation

Authors:
E. B. Saff and V. Totik

Journal:
Trans. Amer. Math. Soc. **316** (1989), 567-593

MSC:
Primary 30E10; Secondary 41A25

DOI:
https://doi.org/10.1090/S0002-9947-1989-0961628-3

MathSciNet review:
961628

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the asymptotic behavior of the polynomials of best uniform approximation to a function that is continuous on a compact set of the complex plane and analytic in the interior of , where has connected complement. For example, we show that for "most" functions , the error does not decrease faster at interior points of than on itself. We also describe the possible limit functions for the normalized error , where , and the possible limit distributions of the extreme points for the error. In contrast to these results, we show that "near best" polynomial approximants to on exist that converge more rapidly at the interior points of .

**[1]**S. Ja. Al′per,*Asymptotic values of best approximation of analytic functions in a complex domain*, Uspehi Mat. Nauk**14**(1959), no. 1 (85), 131–134 (Russian). MR**0104826****[2]**J. M. Anderson and J. Clunie,*Isomorphisms of the disc algebra and inverse Faber sets*, Math. Z.**188**(1985), no. 4, 545–558. MR**774557**, https://doi.org/10.1007/BF01161656**[3]**H.-P. Blatt, E. B. Saff, and M. Simkani,*Jentzsch-Szegő type theorems for the zeros of best approximants*, J. London Math. Soc. (2)**38**(1988), no. 2, 307–316. MR**966302**, https://doi.org/10.1112/jlms/s2-38.2.307**[4]**H.-P. Blatt, E. B. Saff, and V. Totik,*The distribution of extreme points in best complex polynomial approximation*, Constr. Approx.**5**(1989), no. 3, 357–370. MR**996936**, https://doi.org/10.1007/BF01889615**[5]**G. M. Goluzin,*Geometric theory of functions of a complex variable*, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR**0247039****[6]**Kenneth Hoffman,*Banach spaces of analytic functions*, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR**0133008****[7]**M. Ĭ. Kadec′,*On the distribution of points of maximum deviation in the approximation of continuous functions by polynomials*, Uspehi Mat. Nauk**15**(1960), no. 1 (91), 199–202 (Russian). MR**0113079****[8]**Thomas Kövari,*On the order of polynomial approximation for closed Jordan domains*, J. Approximation Theory**5**(1972), 362–373. Collection of articles dedicated to J. L. Walsh on his 75th birthday, IV. MR**0335822****[9]**András Kroó and E. B. Saff,*The density of extreme points in complex polynomial approximation*, Proc. Amer. Math. Soc.**103**(1988), no. 1, 203–209. MR**938669**, https://doi.org/10.1090/S0002-9939-1988-0938669-X**[10]**A. I. Markushevich,*Theory of functions of a complex variable. Vol. III*, Revised English edition, translated and edited by Richard A. Silverman, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR**0215964****[11]**Allan Pinkus,*𝑛-widths in approximation theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 7, Springer-Verlag, Berlin, 1985. MR**774404****[12]**Walter Rudin,*Real and complex analysis*, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. MR**0344043****[13]**E. B. Saff,*A principle of contamination in best polynomial approximation*, Approximation and optimization (Havana, 1987) Lecture Notes in Math., vol. 1354, Springer, Berlin, 1988, pp. 79–97. MR**996661**, https://doi.org/10.1007/BFb0089584**[14]**E. B. Saff and V. Totik,*Limitations of the Carathéodory-Fejér method for polynomial approximation*, J. Approx. Theory**58**(1989), no. 3, 284–296. MR**1012678**, https://doi.org/10.1016/0021-9045(89)90030-0**[15]**-,*Polynomial approximation to piecewise analytic functions*, J. London Math. Soc. (to appear).**[16]**Harold S. Shapiro,*Topics in approximation theory*, Springer-Verlag, Berlin-New York, 1971. With appendices by Jan Boman and Torbjörn Hedberg; Lecture Notes in Math., Vol. 187. MR**0437981****[17]**A. F. Timan,*Theory of approximation of functions of a real variable*, Hindustan, Delhi, 1966.**[18]**Lloyd N. Trefethen,*Near-circularity of the curve in complex Chebyshev approximation*, J. Approx. Theory**31**(1981), no. 4, 344–367. MR**628517**, https://doi.org/10.1016/0021-9045(81)90102-7**[19]**Lloyd N. Trefethen and Martin H. Gutknecht,*The Carathéodory-Fejér method for real rational approximation*, SIAM J. Numer. Anal.**20**(1983), no. 2, 420–436. MR**694530**, https://doi.org/10.1137/0720030**[20]**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, R.I., 1965. MR**0218588**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
30E10,
41A25

Retrieve articles in all journals with MSC: 30E10, 41A25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0961628-3

Article copyright:
© Copyright 1989
American Mathematical Society