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 for generalized polynomials with integral coefficients


Author: M. von Golitschek
Journal: Trans. Amer. Math. Soc. 224 (1976), 417-425
MSC: Primary 41A10
DOI: https://doi.org/10.1090/S0002-9947-1976-0430601-7
MathSciNet review: 0430601
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The classcal Müntz theorem and the so-called Jackson-Müntz theorems concern uniform approximation on [0, 1] by polynomials whose exponents are taken from an increasing sequence of positive real numbers $ \Lambda $. Under mild restrictions on the exponents, the degree of approximation for $ \Lambda $-polynomials with real coefficients is compared with the corresponding degree of approximation when the coefficients are taken from the integers.


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

  • [1] J. Bak and D. J. Newman, Müntz-Jackson theorems in $ {L^p}[0,1]$ and $ C[0,1]$, Amer. J. Math. 94 (1972), 437-457. MR 46 #9605. MR 0310507 (46:9605)
  • [2] S. N. Bernšteĭn, Sobranie sočinenii. I, Izdat. Akad. Nauk SSSR, Moscow, 1952, pp. 468-471, 517-519. MR 14, 2.
  • [3] M. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. 17 (1923), 228-249. MR 1544613
  • [4] Le Baron O. Ferguson and M. von Golitschek, Müntz-Szász theorem with integral coefficients. II, Trans. Amer. Math. Soc. 213 (1975), 115-126. MR 0430619 (55:3624)
  • [5] T. Ganelius and S. Westlund, The degree of approximation in Müntz's theorem, Proc. Internat. Conf. Mathematical Analysis, Jyvaskyla, Finland, 1970.
  • [6] A. O. Gel'fond, On uniform approximations by polynomials with integral rational coefficients, Uspehi Mat. Nauk 10 (1955), no. 1 (63), 41-65. (Russian) MR 17, 30. MR 0070750 (17:30e)
  • [7] M. von Golitschek, Erweiterung der Approximationssätze von Jackson im Sinne von C. Müntz. II, J. Approximation Theory 3 (1970), 72-86. MR 41 #2273. MR 0257623 (41:2273)
  • [8] -, Jackson-Müntz Sätze in der $ {L_p}$-Norm, J. Approximation Theory 7 (1973), 87-106. MR 49 #5641. MR 0340891 (49:5641)
  • [9] E. Hewitt and H. S. Zuckerman, Approximation by polynomials with integral coefficients, a reformulation of the Stone-Weierstrass theorem, Duke Math. J. 26 (1959), 305-324. MR 26 #6656. MR 0149164 (26:6656)
  • [10] S. Kakeya, On approximate polynomials, Tôhoku Math. J. 6 (1914-1915), 182-186.
  • [11] L. Kantorovič, Some remarks on the approximation of functions by means of polynomials with integral coefficients, Izv. Akad. Nauk SSSR, (1931), 1163-1168. (Russian)
  • [12] D. Leviatan, On the Jackson-Müntz theorem, J. Approximation Theory 10 (1974), 1-5. MR 0425428 (54:13383)
  • [13] G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York, 1966. MR 35 #4642; erratum, 36, p. 1567. MR 0213785 (35:4642)
  • [14] C. Müntz, Über den Approximationssatz von Weierstrass, Schwarz-Festschrift, 1914, 302-312.
  • [15] D. J. Newman, A Müntz-Jackson theorem, Amer. J. Math. 87 (1965), 940-944. MR 32 #4429. MR 0186974 (32:4429)
  • [16] Y. Okada, On approximate polynomials with integral coefficients only, Tôhoku Math. J. 23 (1923), 26-35.
  • [17] J. Pál, Zwei kleine Bemerkungen, Tôhoku Math. J. 6 (1914-1915), 42-43.
  • [18] R. M. Trigub, Approximation of functions by polynomials with integer coefficients, Izv. Akad. Nauk SSSR Ser Math. 26 (1962), 261-280. (Russian) MR 25 #373. MR 0136912 (25:373)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 41A10

Retrieve articles in all journals with MSC: 41A10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0430601-7
Keywords: Jackson-Müntz theorem, polynomials with integral coefficients, approximation by polynomials with integral coefficients, degree of approximation
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society