Monotone and comonotone approximation

Authors:
E. Passow and L. Raymon

Journal:
Proc. Amer. Math. Soc. **42** (1974), 390-394

MSC:
Primary 41A50

DOI:
https://doi.org/10.1090/S0002-9939-1974-0336176-9

MathSciNet review:
0336176

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Abstract | References | Similar Articles | Additional Information

Abstract: Jackson type theorems are obtained for monotone and comonotone approximation. Namely

(i) If is a function such that the *k*th difference of *f* is on [*a, b*] then the degree of approximation of *f* by *n*th degree polynomials with *k*th derivative is for any , where is the modulus of continuity of *f* on [*a, b*].

(ii) If is piecewise monotone on [*a, b*] then the degree of approximation of *f* by *n*th degree polynomials comonotone with *f* is for any .

**[1]**G. G. Lorentz and K. L. Zeller,*Degree of approximation by monotone polynomials*. I, J. Approximation Theory**1**(1968), 501-504. MR**39**#699. MR**0239342 (39:699)****[2]**D. J. Newman, E. Passow and L. Raymon,*Piecewise monotone polynomial approximation*, Trans. Amer. Math. Soc.**172**(1972), 465-472. MR**0310506 (46:9604)****[3]**E. Passow and L. Raymon,*Comonotone polynomial approximation*, J. Approximation Theory (to appear). MR**0355422 (50:7896)****[4]**J. A. Roulier,*Monotone approximation of certain classes of functions*, J. Approximation Theory**1**(1968), 319-324. MR**38**#4875. MR**0236580 (38:4875)****[5]**O. Shisha,*Monotone approximation*, Pacific J. Math.**15**(1965), 667-671. MR**32**#2802. MR**0185334 (32:2802)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0336176-9

Keywords:
Monotone approximation,
comonotone approximation,
piecewise monotone approximation,
Jackson theorem

Article copyright:
© Copyright 1974
American Mathematical Society