Monotone and comonotone approximation
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- by E. Passow and L. Raymon PDF
- Proc. Amer. Math. Soc. 42 (1974), 390-394 Request permission
Abstract:
Jackson type theorems are obtained for monotone and comonotone approximation. Namely (i) If $f(x)$ is a function such that the kth difference of f is $\geqq 0$ on [a, b] then the degree of approximation of f by nth degree polynomials with kth derivative $\geqq 0$ is $0[\omega (f;1/{n^{1 - \varepsilon }})]$ for any $\varepsilon > 0$, where $\omega (f;\delta )$ is the modulus of continuity of f on [a, b]. (ii) If $f(x)$ is piecewise monotone on [a, b] then the degree of approximation of f by nth degree polynomials comonotone with f is $0[\omega (f;1/{n^{1 - \varepsilon }})]$ for any $\varepsilon > 0$.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
- D. J. Newman, Eli Passow, and Louis Raymon, Piecewise monotone polynomial approximation, Trans. Amer. Math. Soc. 172 (1972), 465–472. MR 310506, DOI 10.1090/S0002-9947-1972-0310506-9
- Eli Passow and Louis Raymon, Copositive polynomial approximation, J. Approximation Theory 12 (1974), 299–304. MR 355422, DOI 10.1016/0021-9045(74)90071-9
- 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
Additional Information
- © Copyright 1974 American Mathematical Society
- 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