Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Some new multipliers of Fourier series

Author: Martin Buntinas
Journal: Proc. Amer. Math. Soc. 101 (1987), 497-502
MSC: Primary 42A45; Secondary 42A16
MathSciNet review: 908656
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {L^1}$ be the space of all complex-valued $ 2\pi $-periodic integrable functions $ f$ and let $ \widehat{{L^1}}$ be the space of sequences of Fourier coefficients $ \hat f$. A sequence $ \lambda $ is an $ \left( {\widehat{{L^1}} \to \widehat{{L^1}}} \right)$ multiplier if $ \lambda \cdot \hat f = \left( {\lambda \left( n \right)\hat f\left( n \right)} \right)$ belongs to $ \widehat{{L^1}}$ for every $ f$ in $ {L^1}$. The space of even sequences of bounded variation is defined by $ b\upsilon = \left\{ {\lambda \left\vert {{\lambda _n} = {\lambda _{ - n}},\sum... ...+ {{\sup }_n}\left\vert {{\lambda _n}} \right\vert < \infty } \right.} \right\}$, where $ \Delta {\lambda _k} = {\lambda _k} - {\lambda _{k + 1}}$ and the space of even bounded quasiconvex sequences is defined by $ q = \left\{ {\lambda \left\vert {{\lambda _n} = {\lambda _{ - n}},\sum\nolimit... ...{{\sup }_n}\left\vert {{\lambda _n}} \right\vert < \infty } } \right.} \right\}$, where $ {\Delta ^2}{\lambda _k} = \Delta {\lambda _k} - \Delta {\lambda _{k + 1}}$. It is well known that $ q \subset b\upsilon $ and $ q \subset \left( {\widehat{{L^1}} \to \widehat{{L^1}}} \right)$ but $ b\upsilon \not\subset \left( {\widehat{{L^1}} \to \widehat{{L^1}}} \right)$. This result is significantly improved by finding an increasing family of sequence spaces $ d{\upsilon_p}$ between $ q$ and $ b\upsilon$ which are $ \left( {\widehat{{L^1}} \to \widehat{{L^1}}} \right)$ multipliers. Since the $ \left( {\widehat{{L^1}} \to \widehat{{L^1}}} \right)$ multipliers are the $ 2\pi $-periodic measures, this result gives sufficient conditions for a sequence to be the Fourier coefficients of a measure.

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

  • [1] N. Bary, A treatise on trigonometric series, vols. 1 and 2, Pergamon Press, New York, 1964.
  • [2] Martin Buntinas, Convergent and bounded Cesàro sections in 𝐹𝐾-spaces, Math. Z. 121 (1971), 191–200. MR 0295020
  • [3] R. E. Edwards, Fourier series: A modern introduction, vols. 1 and 2, Holt, Rinehart and Winston, New York, 1967.
  • [4] G. A. Fomin, A class of trigonometric series, Mat. Zametki 23 (1978), no. 2, 213–222 (Russian). MR 0487218
  • [5] W. Orlicz, Über $ k$-fach monotone Folgen, Studia Math. 6 (1936), 149-159.
  • [6] A. Zygmund, Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968. MR 0236587

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42A45, 42A16

Retrieve articles in all journals with MSC: 42A45, 42A16

Additional Information

Article copyright: © Copyright 1987 American Mathematical Society