ArticleOriginal scientific text

Title

A counterexample in comonotone approximation in Lp space

Authors 1, 2

Affiliations

  1. Department of Mathematics, Hangzhou University, Hangzhou, Zhejiang, P.R. China
  2. Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G.2G1

Abstract

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 fC-11k, 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 en(k)(f)p is the best approximation of degree n to f in Lp 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 Lp space for 1 < p ≤ ∞.

Bibliography

  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 En(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 Lp for 0
  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 Lp, !$!0
  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 Lp 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.
Pages:
265-274
Main language of publication
English
Received
1992-05-11
Published
1993
Exact and natural sciences