Piecewise monotone polynomial approximation

Newman, D. J.; Passow, Eli; Raymon, Louis

doi:10.1090/S0002-9947-1972-0310506-9

Piecewise monotone polynomial approximation
HTML articles powered by AMS MathViewer

by D. J. Newman, Eli Passow and Louis Raymon

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

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

PDF | Request permission

Abstract:

Given a real function $f$ satisfying a Lipschitz condition of order 1 on $[a,b]$, there exists a sequence of approximating polynomials $\{ {P_n}\}$ such that the sequence ${E_n} = ||{P_n} - f||$ (sup norm) has order of magnitude $1/n$ (D. Jackson). We investigate the possibility of selecting polynomials ${P_n}$ having the same local monotonicity as $f$ without affecting the order of magnitude of the error. In particular, we establish that if $f$ has a finite number of maxima and minima on $[a,b]$ and $S$ is a closed subset of $[a,b]$ not containing any of the extreme points of $f$, then there is a sequence of polynomials ${P_n}$ such that ${E_n}$ has order of magnitude $1/n$ and such that for $n$ sufficiently large ${P_n}$ and $f$ have the same monotonicity at each point of $S$. The methods are classical.

References

G. G. Lorentz and K. L. Zeller, Degree of approximation by monotone polynomials. I, J. Approximation Theory 1 (1968), 501–504. MR 239342, DOI 10.1016/0021-9045(68)90039-7
Günter Meinardus, Approximation of functions: Theory and numerical methods, Expanded translation of the German edition, Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. Translated by Larry L. Schumaker. MR 0217482
Eli Passow, Another proof of Jackson’s theorem, J. Approximation Theory 3 (1970), 146–148. MR 262748, DOI 10.1016/0021-9045(70)90022-5
John A. Roulier, Monotone approximation of certain classes of functions, J. Approximation Theory 1 (1968), 319–324. MR 236580, DOI 10.1016/0021-9045(68)90009-9
O. Shisha, Monotone approximation, Pacific J. Math. 15 (1965), 667–671. MR 185334
W. Wolibner, Sur un polynôme d’interpolation, Colloq. Math. 2 (1951), 136–137 (French). MR 43946, DOI 10.4064/cm-2-2-136-137
Sam W. Young, Piecewise monotone polynomial interpolation, Bull. Amer. Math. Soc. 73 (1967), 642–643. MR 212455, DOI 10.1090/S0002-9904-1967-11806-8