Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Fourier series of functions of $ \Lambda $-bounded variation


Author: Daniel Waterman
Journal: Proc. Amer. Math. Soc. 74 (1979), 119-123
MSC: Primary 42A16; Secondary 42A20
MathSciNet review: 521884
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the Fourier coefficients of functions of $ \Lambda $-bounded variation, $ \Lambda = \{ {\lambda _n}\} $, are $ O({\lambda _n}/n)$. This was known for $ {\lambda _n} = {n^{\beta + 1}}, - 1 \leqslant \beta < 0$. The classes L and HBV are shown to be complementary, but L and $ \Lambda {\text{BV}}$ are not complementary if $ \Lambda {\text{BV}}$ is not contained in HBV. The partial sums of the Fourier series of a function of harmonic bounded variation are shown to be uniformly bounded and a theorem analogous to that of Dirichlet is shown for this class of functions without recourse to the Lebesgue test.


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

  • [1] Daniel Waterman, On convergence of Fourier series of functions of generalized bounded variation, Studia Math. 44 (1972), 107–117. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity. II. MR 0310525 (46 #9623)
  • [2] -, On the summability of Fourier series of functions of $ \Lambda $-bounded variation, Studia Math. 55 (1976), 87-95.
  • [3] Daniel Waterman, On 𝐿-bounded variation, Studia Math. 57 (1976), no. 1, 33–45. MR 0417355 (54 #5408)
  • [4] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776 (21 #6498)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42A16, 42A20

Retrieve articles in all journals with MSC: 42A16, 42A20


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0521884-1
PII: S 0002-9939(1979)0521884-1
Article copyright: © Copyright 1979 American Mathematical Society