Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Generalized convex functions and best $ L\sb p$ approximation


Authors: Ronald M. Mathsen and Vasant A. Ubhaya
Journal: Proc. Amer. Math. Soc. 114 (1992), 733-740
MSC: Primary 26A51; Secondary 41A50
DOI: https://doi.org/10.1090/S0002-9939-1992-1088444-4
MathSciNet review: 1088444
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Some properties of generalized convex functions significant to approximation theory are obtained. The existence of a best $ {L_p}$ approximation $ (1 \leq p \leq \infty )$ from subsets of these functions is established under certain conditions. Special cases of these functions include $ n$-convex functions which are much investigated in the literature.


References [Enhancements On Off] (What's this?)

  • [1] E. F. Beckenbach, Generalized convex functions, Bull. Amer. Math. Soc. 43 (1937), 363-371. MR 1563543
  • [2] E. F. Beckenbach and R. H. Bing, On generalized convex functions, Trans. Amer. Math. Soc. 58 (1945), 220-230. MR 0013169 (7:116c)
  • [3] P. S. Bullen, A criterion for $ n$-convexity, Pacific J. Math. 36 (1971), 81-98. MR 0274681 (43:443)
  • [4] P. Hartman, Unrestricted $ n$-parameter families, Rend. Circ. Mat. Palermo (2) 7 (1958), 123-142. MR 0105470 (21:4211)
  • [5] R. Huotari, D. Legg, A. D. Meyerowitz, and D. Townsend, The natural best $ {L_1}$-approximation by nondecreasing functions, J. Approx. Theory 52 (1988), 132-140. MR 929299 (89b:41025)
  • [6] R. Huotari, R. Legg, and D. Townsend, Existence of best $ n$-convex approximants in $ {L_1}$, Approx. Theory Appl. 5 (1989), 51-57. MR 1023960 (90m:41031)
  • [7] D. Landers and L. Rogge, Isotonic approximation in $ {L_s}$, J. Approx. Theory 31 (1981), 199-223. MR 624009 (82j:60055)
  • [8] S. Karlin and W. J. Studden, Tchebycheff systems: with applications in analysis and statistics, Interscience, New York, 1966. MR 0204922 (34:4757)
  • [9] J. H. B. Kemperman, On the regularity of generalized convex functions, Trans. Amer. Math. Soc. 135 (1969), 69-93. MR 0265531 (42:440)
  • [10] M. G. Krein and A. A. Nudel'man, The Markov moment problem and extremal problems, Trans. Math. Monographs, vol. 50, Amer. Math. Soc., Providence, RI, 1977. MR 0458081 (56:16284)
  • [11] J. T. Lewis and O. Shisha, $ {L_p}$ convergence of monotone functions and their uniform convergence, J. Approx. Theory 14 (1975), 281-284. MR 0382581 (52:3464)
  • [12] R. M. Mathsen, $ \lambda (n)$-convex functions, Rocky Mountain J. Math. 2 (1972), 31-43. MR 0294581 (45:3651)
  • [13] -, Hereditary $ \lambda (n,k)$-families and generalized convexity of functions, Rocky Mountain J. Math. 12 (1982), 753-756. MR 683867 (84c:26015)
  • [14] E. Moldovan, Sur une généralisation des fonctions convexes, Matematica (Cluj) (2) 1 (1959), 49-80. MR 0136905 (25:366)
  • [15] T. Popoviciu, Les fonctions convexes, Hermann, Paris, 1944. MR 0018705 (8:319a)
  • [16] A. W. Roberts and D. E. Varberg, Convex functions, Academic Press, New York, 1973. MR 0442824 (56:1201)
  • [17] R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, NJ, 1970.
  • [18] J. J. Swetits, S. E. Weinstein, and Y. Xu, On the characterization and computation of best monotone approximation in $ {L_p}[0,1]$ for $ 1 \leq p < \infty $, J. Approx. Theory 60 (1990), 58-68. MR 1028894 (90j:41047)
  • [19] J. J. Swetits and S. E. Weinstein, Construction of best monotone approximation on $ {L_p}[0,1]$, J. Approx. Theory 61 (1990), 118-130. MR 1047153 (91a:41033)
  • [20] J. J. Swetits, S. E. Weinstein, and Y. Xu, Approximation in $ {L_p}[0,1]$ by $ n$-convex functions, Numer. Funct. Anal. Optim. 11 (1990), 167-179. MR 1058784 (91c:41051)
  • [21] L. Tornheim, On $ n$-parameter families of functions and associated convex functions, Trans. Amer. Math. Soc. 69 (1950), 457-467. MR 0038383 (12:395d)
  • [22] -, Approximation by families of functions, Trans, Amer. Math. Soc. 7 (1956), 641-643. MR 0079671 (18:125f)
  • [23] V. A. Ubhaya, An $ O(n)$ algorithm for discrete $ n$-point convex approximation with applications to continuous case, J. Math. Anal. Appl. 72 (1979), 338-354. MR 552341 (81b:41071)
  • [24] -, $ {L_p}$ approximation from nonconvex subsets of special classes of functions, J. Approx. Theory 57 (1989), 223-238. MR 993644 (91a:41022)
  • [25] D. Zwick, Existence of best $ n$-convex approximations, Proc. Amer. Math. Soc. 97 (1986), 273-276. MR 835879 (87j:41052)
  • [26] -, Best $ {L_1}$-approximation by generalized convex functions, J. Approx. Theory 59 (1989), 116-123.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A51, 41A50

Retrieve articles in all journals with MSC: 26A51, 41A50


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1088444-4
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society