Complex approximation of real functions by reciprocals of polynomials

Author:
Daniel Wulbert

Journal:
Trans. Amer. Math. Soc. **316** (1989), 635-652

MSC:
Primary 41A20

DOI:
https://doi.org/10.1090/S0002-9947-1989-0967318-5

MathSciNet review:
967318

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Characterizations are given for local and global best rational approximations to a real function. The characterizations are specialized to reciprocals of polynomials, where they are used to settle some conjectures and questions.

**[1]**D. Braess,*Nonlinear approximation theory*, Springer-Verlag, Berlin, 1986. MR**866667 (88e:41002)****[2]**E. W. Cheney,*Introduction to approximation theory*, McGraw-Hill, New York, 1966. MR**0222517 (36:5568)****[3]**M. H. Gutknecht and L. N. Trefetha,*Real and complex Chebyshev approximation on the unit disk and interval*, Bull. Amer. Math. Soc.**8**(1983), 455-459. MR**693961 (84d:30068)****[4]**-,*Real vs complex rational Chebyshev approximation on an interval*, Trans. Amer. Math. Soc.**280**(1983), 555-561. MR**716837 (85h:41045)****[5]**D. J. Newman,*Approximation with rational functions*, CBMS Regional Conf. Ser. in Math., Amer. Math. Soc., Providence, R.I., 1979. MR**539314 (84k:41019)****[6]**T. J. Rivlin and H. S. Shapiro,*A unified approach to certain problems of approximation and minimization*, J. Soc. Indust. Appl. Math.**9**(1961), 670-699. MR**0133636 (24:A3462)****[7]**A. Ruttan,*A characterization of best complex rational approximants in a fundamental case*, Constructive Approximation**1**(1985), 287-296. MR**891533 (88h:30060)****[8]**-,*The length of the alternation set as a factor in determining when a best real rational approximation is also a best complex rational approximation*, J. Approx. Theory**31**(1981), 230-243. MR**624011 (84d:41026)****[9]**J. L. Walsh,*The existence of rational functions of best approximation*, Trans. Amer. Math. Soc.**33**(1931), 668-689. MR**1501609****[10]**D. E. Wulbert,*The rational approximation of real functions*, Amer. J. Math.**100**(1978), 1281-1317. MR**522701 (80i:41012)****[11]**-,*The characterization of complex rational approximations*, Illinois J. Math.**24**(1980), 140-155. MR**550656 (81b:30073)****[12]**-,*Local best rational approximations to continuous functions and the rays they emanate*, J. Approx. Theory**52**(1988), 350-358. MR**934799 (89h:41039)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
41A20

Retrieve articles in all journals with MSC: 41A20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0967318-5

Article copyright:
© Copyright 1989
American Mathematical Society