Nearly comonotone approximation

Author:
John A. Roulier

Journal:
Proc. Amer. Math. Soc. **47** (1975), 84-88

MSC:
Primary 41A25

MathSciNet review:
0364967

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Abstract: This paper obtains estimates on the degree of nearly comonotone approximation which extend and improve the estimate obtained by Newman, Passow, and Raymon. In particular, the restriction that is removed, and estimates for the degree of nearly comonotone approximation are obtained for all proper piecewise monotone functions. It is also shown that if exists and is continuous on the interval, then the ordinary polynomials of best approximation form a nearly comonotone sequence.

**[1]**G. G. Lorentz,*Monotone approximation*, Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), Academic Press, New York, 1972, pp. 201–215. MR**0346375****[2]**G. G. Lorentz and K. L. Zeller,*Degree of approximation by monotone polynomials. I*, J. Approximation Theory**1**(1968), 501–504. MR**0239342****[3]**G. G. Lorentz and K. L. Zeller,*Degree of approximation by monotone polynomials. II*, J. Approximation Theory**2**(1969), 265–269. MR**0244677****[4]**D. J. Newman, Eli Passow, and Louis Raymon,*Piecewise monotone polynomial approximation*, Trans. Amer. Math. Soc.**172**(1972), 465–472. MR**0310506**, 10.1090/S0002-9947-1972-0310506-9**[5]**John A. Roulier,*Monotone approximation of certain classes of functions*, J. Approximation Theory**1**(1968), 319–324. MR**0236580****[6]**J. A. Roulier,*Monotone and weighted approximation*, Doctoral Dissertation, Syracuse University, Syracuse, N. Y., 1968.**[7]**O. Shisha,*Monotone approximation*, Pacific J. Math.**15**(1965), 667–671. MR**0185334**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1975-0364967-8

Article copyright:
© Copyright 1975
American Mathematical Society