Piecewise monotone polynomial approximation
Authors:
D. J. Newman, Eli Passow and Louis Raymon
Journal:
Trans. Amer. Math. Soc. 172 (1972), 465472
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.
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 G. Meinardus, Approximation of functions: Theory and numerical methods, Springer, Berlin, 1964; English transl., SpringerTracts in Natural Philosophy, vol 13, SpringerVerlag, New York, 1967. MR 31 #547; MR 36 #571. MR 0217482 (36:571)
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 Eli Passow, Another proof of Jackson's theorem, J. Approximation Theory 3 (1970), 146148. MR 41 #7353. MR 0262748 (41:7353)
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 John A. Roulier, Monotone approximation of certain classes of functions, J. Approximation Theory 1 (1968), 319324. MR 38 #4875. MR 0236580 (38:4875)
 [5]
 O. Shisha, Monotone approximation, Pacific J. Math. 15 (1965), 667671. MR 32 #2802. MR 0185334 (32:2802)
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 W. Wolibner, Sur un polynôme d'interpolation, Colloq. Math. 2 (1951), 136137. MR 13, 343. MR 0043946 (13:343e)
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 S. W. Young, Piecewise monotone polynomial interpolation, Bull. Amer. Math. Soc. 73 (1967), 642643. MR 35 #3326. MR 0212455 (35:3326)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197203105069
PII:
S 00029947(1972)03105069
Keywords:
Monotone approximation,
piecewise monotone approximation,
Jackson kernel,
Jackson's Theorem
Article copyright:
© Copyright 1972
American Mathematical Society
