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On the degree of approximation to discontinuous functions by trigonometric sums

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Rendiconti del Circolo Matematico di Palermo (1884-1940)

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References

  1. See e. g.D. Jackson,On approximation by trigonometric sums and polynomials [Transactions of the American Mathematical Society, vol. XIII (1912), pp. 491–515], p. 492.

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  2. Byoscillation is meant the difference between the upper and lower limits off(x) in the interval.

  3. D. Jackson, loc. cit. I), p. 492.

  4. The definition adopted for φ(x) at the points of discontinuity off(x) is immaterial.

  5. D. Jackson, loe. cit. I), p. 492.

  6. This language is not intended to imply that the coefficient of xn is necessarily different from zero.

  7. See e. g.D. Jackson, loc. cit. I), p. 494. His assumption that the function represented is continuous, is obviously unnecessary.

  8. D. Jackson, loc. cit. I), p. 492.

  9. The fact that the order of the trigonometric sumI m (x) is notm, but 2(m-1), is obviously inessential.

  10. For the case thatf(x) is everywhere continuous, see S. Bernstein,Sur ľordre de la meilleure approximation des fonctions continues par des polyn0mes de degré donné [Mémoires couronnés et autres mémoires publiés par ľAcadémie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique (Bruxelles), Series II, Vol. IV (1912), fasc. I, pp. 1–104], PP– 88, 89.

  11. For proof of these statements seeM. Bôcher,Introduction to the Theory of Fourier’sSeries [Annals of Mathematics, vol. VII (1906), pp. 81–152], pp. 123 ff.

  12. For proof of this fact see the corresponding passage in the proof of Theorem I.

  13. SeeH. Lebesgue,Sur la représentation trigonométrique approchée des fonctions satisfaisant à une condition de Lipschitz [Bulletin de la Société Mathématique de France, vol. XXXVIII (1910), pp. 184–210], p. 201;D. Jackson, loc. cit. I), p. 502.

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  14. S. Bernstein, loc. cit. 10), pp. 98–100.

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  1. Cambridge, Mass.

    Charles E. Wilder

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  1. Charles E. Wilder
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Wilder, C.E. On the degree of approximation to discontinuous functions by trigonometric sums. Rend. Circ. Mat. Palermo 39, 345–361 (1915). https://doi.org/10.1007/BF03015986

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