Piecewise monotone polynomial approximation

Authors:
D. J. Newman, Eli Passow and Louis Raymon

Journal:
Trans. Amer. Math. Soc. **172** (1972), 465-472

MSC:
Primary 41A25; Secondary 41A10

MathSciNet review:
0310506

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Abstract: Given a real function satisfying a Lipschitz condition of order 1 on , there exists a sequence of approximating polynomials such that the sequence (sup norm) has order of magnitude (D. Jackson). We investigate the possibility of selecting polynomials having the same local monotonicity as without affecting the order of magnitude of the error. In particular, we establish that if has a finite number of maxima and minima on and is a closed subset of not containing any of the extreme points of , then there is a sequence of polynomials such that has order of magnitude and such that for sufficiently large and have the same monotonicity at each point of . The methods are classical.

**[1]**G. G. Lorentz and K. L. Zeller,*Degree of approximation by monotone polynomials. I*, J. Approximation Theory**1**(1968), 501–504. MR**0239342****[2]**Günter Meinardus,*Approximation of functions: Theory and numerical methods*, Expanded translation of the German edition. Translated by Larry L. Schumaker. Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. MR**0217482****[3]**Eli Passow,*Another proof of Jackson’s theorem*, J. Approximation Theory**3**(1970), 146–148. MR**0262748****[4]**John A. Roulier,*Monotone approximation of certain classes of functions*, J. Approximation Theory**1**(1968), 319–324. MR**0236580****[5]**O. Shisha,*Monotone approximation*, Pacific J. Math.**15**(1965), 667–671. MR**0185334****[6]**W. Wolibner,*Sur un polynôme d’interpolation*, Colloquium Math.**2**(1951), 136–137 (French). MR**0043946****[7]**Sam W. Young,*Piecewise monotone polynomial interpolation*, Bull. Amer. Math. Soc.**73**(1967), 642–643. MR**0212455**, 10.1090/S0002-9904-1967-11806-8

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DOI:
https://doi.org/10.1090/S0002-9947-1972-0310506-9

Keywords:
Monotone approximation,
piecewise monotone approximation,
Jackson kernel,
Jackson's Theorem

Article copyright:
© Copyright 1972
American Mathematical Society