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)

 
 

 

On the arithmetic mean of Fourier-Stieltjes coefficients


Author: Constantine Georgakis
Journal: Proc. Amer. Math. Soc. 33 (1972), 477-484
MSC: Primary 42A16
DOI: https://doi.org/10.1090/S0002-9939-1972-0298319-3
MathSciNet review: 0298319
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \{ {a_n}\} _{n = 0}^\infty $ be the cosine Fourier-Stieltjes coefficients of the Borel measure $ \mu $ and $ \{ {a_0},({a_1} + \cdots + {a_n})/n\} _{n = 1}^\infty = \{ {(Ta)_n}\} _{n = 0}^\infty $ be the sequence of their arithmetic means. Then $ \sum\nolimits_{n = 0}^\infty {{{(Ta)}_n}\cos nx} $ is a Fourier-Stieltjes series. Moreover, (a) $ \sum\nolimits_{n = 0}^\infty {{{(Ta)}_n}\cos nx} $ is a Fourier series if and only if $ {(Ta)_n} \to 0$ at infinity or, equivalently, the measure $ \mu $ is continuous at the origin, (b) $ \sum\nolimits_{n = 1}^\infty {{{(Ta)}_n}\sin nx} $ is a Fourier series if and only if the function $ {x^{ - 1}}\mu ([0,x))$ is in $ {L^1}[0,\pi ]$. These results form the best possible analogue of a theorem of G. Goes, concerning arithmetic means of Fourier-Stieltjes sine coefficients, and improve considerably the theorems of L. Fejér and N. Wiener on the inversion and quadratic variation of Fourier-Stieltjes coefficients.


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

  • [1] R. P. Boas, Jr. and S. Izumi, Absolute convergence of some trigonometric series. I, J. Indian Math. Soc. 24 (I960), 191-210. MR 23 #A1198. MR 0123877 (23:A1198)
  • [2] L. Fejér, Über die Bestimung des Sprunges einer Funktion aus ihrer Fourierreihe, J. Reine Angew. Math. 142 (1913) 165-168.
  • [3] M. Girault, Transformation de fonctions caractéristiques par intégration, C. R. Acad. Sci. Paris 238 (1954), 2223-2224. MR 16, 52. MR 0062977 (16:52f)
  • [4] G. Goes, Arithmetic means of Fourier-Stieltjes-sine-coejficients, Proc. Amer. Math. Soc. 14 (1963), 10-11. MR 26 #2791. MR 0145258 (26:2791)
  • [5] R. R. Goldberg, Averages of Fourier coefficients, Pacific J. Math. 9 (1959), 695-699. MR 22 #2842. MR 0111984 (22:2842)
  • [6] G. H. Hardy, Notes on some points in the integral calculus, Messenger Math. 58 (1929), 50-52.
  • [7] -, On a theorem of Palev and Wiener, Proc. Cambridge Philos. Soc. 33 (1937), 1-5.
  • [8] M. Izumi and S. Izumi, On a Hardy's theorem, Proc. Japan Acad. 44 (1968), 418-423. MR 38 #2519. MR 0234201 (38:2519)
  • [9] M. Kinukawa and S. Igari, Transformations of conjugate functions, Tôhoku Math. J. (2) 13 (1961), 274-280. MR 26 #2829. MR 0145298 (26:2829)
  • [10] C. T. Loo, Note on the properties of Fourier coefficients, Amer. J. Math. 71 (1949), 269-282. MR 10, 603. MR 0029440 (10:603e)
  • [11] N. Wiener, The quadratic variation of a function and its Fourier constants, Mass. J. Math. 3 (1924), 73-94.
  • [12] A. Zygmund, Some points in the theory of trigonometric and power series, Trans. Amer. Math. Soc. 36 (1934), 586-617, p. 615. MR 1501757
  • [13] -, Trigonometrical series. Vol. I, 2nd ed., Cambridge Univ. Press, New York, 1959. MR 21 #6498.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42A16

Retrieve articles in all journals with MSC: 42A16


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0298319-3
Keywords: Fourier-Stieltjes coefficients, Borel measure, discrete measure, inversion, arithmetic mean, Fourier coefficients, conjugate function, conjugate series
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society