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## Colloquium Mathematicum

1993 | 64 | 2 | 265-274
Tytuł artykułu

### A counterexample in comonotone approximation in $L^p$ space

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Refining the idea used in [24] and employing very careful computation, the present paper shows that for 0 < p ≤ ∞ and k ≥ 1, there exists a function $f ∈ C_{[-1,1]}^k$, with $f^{(k)}(x)≥ 0$ for x ∈ [0,1] and $f^{(k)}(x) ≤ 0$ for x ∈ [-1,0], such that lim sup_{n→∞} (e_n^{(k)}(f)_p) / (ω_{k+2+[1/p]}(f,n^{-1})_{p}) = + ∞ where $e_n^{(k)}(f)_p$ is the best approximation of degree n to f in $L^p$ by polynomials which are comonotone with f, that is, polynomials P so that $P^{(k)}(x)f^{(k)}(x) ≥ 0$ for all x ∈ [-1,1]. This theorem, which is a particular case of a more general one, gives a complete solution to the converse result in comonotone approximation in $L^p$ space for 1 < p ≤ ∞.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
265-274
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-05-11
Twórcy
autor
• Department of Mathematics, Hangzhou University, Hangzhou, Zhejiang, P.R. China
autor
• Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G.2G1
Bibliografia
• [1] R. K. Beatson, The degree of monotone approximation, Pacific J. Math. 74 (1978), 5-14.
• [2] R. K. Beatson and D. Leviatan, On comonotone approximation, Canad. Math. Bull. 26 (1983), 220-224.
• [3] R. A. DeVore, Degree of approximation, in: Approximation Theory II, Academic Press, New York 1976, 117-162.
• [4] R. A. DeVore, Monotone approximation by polynomials, SIAM J. Math. Anal. 8 (1977), 906-921.
• [5] R. A. DeVore and X. M. Yu, Pointwise estimates for monotone polynomial approximation, Constr. Approx. 1 (1985), 323-331.
• [6] M. Hasson, Functions f for which $E_n(f)$ is exactly of the order $n^-1$, in: Approximation Theory III, Academic Press, New York 1980, 491-494.
• [7] G. L. Iliev, Exact estimates for partially monotone approximation, Anal. Math. 4 (1978), 181-197.
• [8] D. Leviatan, The behavior of the derivatives of the algebraic polynomials of best approximation, J. Approx. Theory 35 (1982), 169-176.
• [9] D. Leviatan, Monotone and comonotone polynomial approximation revisited, ibid. 53 (1988), 1-16.
• [10] D. Leviatan, Monotone polynomial approximation, Rocky Mountain J. Math. 19 (1989), 231-241.
• [11] G. G. Lorentz, Monotone approximation, in: Inequalities III, Academic Press, New York 1972, 201-215.
• [12] G. G. Lorentz and K. Zeller, Degree of approximation by monotone polynomials I, J. Approx. Theory 1 (1968), 501-504.
• [13] G. G. Lorentz and K. Zeller, Degree of approximation by monotone polynomials II, ibid. 2 (1969), 265-269.
• [14] P. G. Nevai, Bernstein's inequality in $L^p$ for 0<p<1, ibid. 27 (1979), 239-243.
• [15] D. J. Newman, Efficient comonotone approximation, ibid. 25 (1979), 189-192.
• [16] E. Passow and L. Raymon, Monotone and comonotone approximation, Proc. Amer. Math. Soc. 42 (1974), 390-394.
• [17] E. Passow, L. Raymon and J. A. Roulier, Comonotone polynomial approximation, J. Approx. Theory 11 (1974), 221-224.
• [18] J. A. Roulier, Monotone approximation of certain classes of functions, ibid. 1 (1968), 319-324.
• [19] J. A. Roulier, Some remarks on the degree of monotone approximation, ibid. 14 (1975), 225-229.
• [20] O. Shisha, Monotone approximation, Pacific J. Math. 15 (1965), 667-671.
• [21] A. S. Shvedov, Jackson's theorem in $L^p$, $0<p<1$, for algebraic polynomials, and orders of comonotone approximation, Math. Notes 25 (1979), 57-65.
• [22] A. S. Shvedov, Orders of coapproximation of functions by algebraic polynomials, ibid. 29 (1981), 63-70.
• [23] X. Wu and S. P. Zhou, A problem on coapproximation of functions by algebraic polynomials, in: Progress in Approximation Theory, P. Nevai and A. Pinkus (eds.), Academic Press, New York 1991, 857-866.
• [24] X. Wu and S. P. Zhou, On a counterexample in monotone approximation, J. Approx. Theory 69 (1992), 205-211.
• [25] X. M. Yu, Pointwise estimates for convex polynomial approximation, Approx. Theory Appl. 1 (4) (1985), 65-74.
• [26] X. M. Yu, Degree of comonotone polynomial approximation, ibid. 4 (3) (1988), 73-78; MR 90c:41042.
• [27] X. M. Yu and Y. P. Ma, Generalized monotone approximation in $L_p$ space, Acta Math. Sinica (N.S.) 5(1989), 48-56; MR 90d:41014.
• [28] S. P. Zhou, A proof of a theorem of Hasson, Vestnik Beloruss. Gos. Univ. Ser. I Fiz. Mat. Mekh. 1988 (3), 56-58, 79 (in Russian); MR 89m:41006.
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