Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The degree of approximation by Chebyshevian splines


Authors: R. DeVore and F. Richards
Journal: Trans. Amer. Math. Soc. 181 (1973), 401-418
MSC: Primary 41A15
DOI: https://doi.org/10.1090/S0002-9947-1973-0336160-9
MathSciNet review: 0336160
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies the connections between the smoothness of a function and its degree of approximation by Chebyshevian splines. This is accomplished by proving companion direct and inverse theorems which give a characterization of smoothness in terms of degree of approximation. A determination of the saturation properties is included.


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

  • [1] G. Butler and F. Richards, An $ {L_p}$-saturation theorem for splines, Canad. J. Math. (to appear). MR 0308658 (46:7772)
  • [2] G. Freud and V. Popov, On approximation by spline functions, Proc. Conference on Constructive Theory of Functions, Budapest, 1968, pp. 163-172. MR 0397246 (53:1105)
  • [3] D. Gaier, Saturation bei Spline-Approximation und Quadratur, Numer. Math. 16 (1970), 129-140. MR 42 #8692. MR 0273816 (42:8692)
  • [4] S. Karlin, Total positivity. Vol. I, Stanford Univ. Press, Stanford, Calif., 1968. MR 37 #5667. MR 0230102 (37:5667)
  • [5] S. Karlin and W. J. Studden, Tchebyeheff systems: With applications in analysis and statistics, Pure and Appl. Math., vol. 15, Interscience, New York, 1966. MR 34 #4757. MR 0204922 (34:4757)
  • [6] G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York, 1966. MR 35 #4642. MR 0213785 (35:4642)
  • [7] G. Meinardus, Approximation von Funktionen ud ihre numerische Behandlung, Springer Tracts in Natural Philosophy, vol. 4, Springer-Verlag, Berlin and New York, 1964. MR 31 #547. MR 0176272 (31:547)
  • [8] V. A. Popov and Bl. K. Sendov, The classes that are characterized by the best approximation by spline functions, Mat. Zametki 8 (1970), 137-148 = Math. Notes 8 (1970), 550-557. MR 43 #5224. MR 0279502 (43:5224)
  • [9] F. Richards, On the saturation class for spline functions, Proc. Amer. Math. Soc. 33 (1972), 471-476. MR 0294958 (45:4026)
  • [10] K. Scherer,, On the best approximation of continuous functions by splines, SIAM J. Numer. Anal. 7 (1970), 418-423. MR 42 #2634. MR 0267732 (42:2634)
  • [11] A. F. Timan, Theory of approximation of functions of a real variable, Fizmatgiz, Moscow, 1960; English transl., Internat. Series of Monographs in Pure and Appl. Math., vol. 34, Macmillan, New York, 1963. MR 22 #8257; MR 33 #465. MR 0117478 (22:8257)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 41A15

Retrieve articles in all journals with MSC: 41A15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0336160-9
Keywords: Splines, degree of approximation, Chebyshev system, saturation, inverse theorems
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society