Pointwise estimates for convex polynomial approximation

Author:
D. Leviatan

Journal:
Proc. Amer. Math. Soc. **98** (1986), 471-474

MSC:
Primary 41A10; Secondary 26A51, 41A25

DOI:
https://doi.org/10.1090/S0002-9939-1986-0857944-9

MathSciNet review:
857944

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Abstract | References | Similar Articles | Additional Information

Abstract: For a convex function we construct a sequence of convex polynomials of degree not exceeding such that . If in addition is monotone it follows that the polynomials are also monotone thus providing simultaneous monotone and convex approximation.

**[1]**R. A. DeVore,*Monotone approximation by polynomials*, SIAM J. Math. Anal.**8**(1977), 906-921. MR**0510582 (58:23252)****[2]**R. A. DeVore and X. M. Yu,*Pointwise estimates for monotone polynomial approximation*, Constr. Approx.**1**(1985), 323-331. MR**891762 (88h:41010)****[3]**Z. Ditzian,*On interpolation of**and weighted Sobolev spaces*, Pacific J. Math.**90**(1980), 307-323. MR**600633 (82c:46038)****[4]**-,*Polynomials of best approximation in*, Israel J. Math.**52**(1985), 341-354. MR**829363 (87g:41012)****[5]**Z. Ditzian and V. Totik,*Moduli of smoothness*, unpublished manuscript.**[6]**D. Leviatan,*Monotone and comonotone approximation revisited*, J. Approx. Theory (to appear). MR**937138 (89h:41017)****[7]**G. G. Lorentz,*Monotone approximation*, Inequalities. III (O. Shisha, ed.), Academic Press, New York, 1972, pp. 201-215. MR**0346375 (49:11100)****[8]**G. G. Lorentz and K. Zeller,*Degree of approximation by monotone polynomials*. I, J. Approx. Theory**1**(1968), 501-504. MR**0239342 (39:699)****[9]**A. S. Shvedov,*Jackson's theorem in*, ,*for algebraic polynomials and orders of comonotone approximations*, Mat. Zametki**25**(1979), 107-117; English transl., Math. Notes**25**(1979), 57-63. MR**527004 (81c:41017)****[10]**-,*Orders of coapproximation of functions by algebraic polynomials*, Mat. Zametki**30**(1981), 117-130; English transl., Math. Notes**30**(1981), 63-70. MR**604156 (82c:41009)****[11]**V. Totik,*An interpolation theorem and its applications to positive operators*, Pacific J. Math.**111**(1984), 447-481. MR**734866 (86g:41033)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1986-0857944-9

Keywords:
Degree of convex polynomial approximation,
Jackson-Timan-Teljakowskiĭ type estimates,
moduli of smoothness,
the Peetre kernel

Article copyright:
© Copyright 1986
American Mathematical Society