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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
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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?)

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Article copyright: © Copyright 1987 American Mathematical Society

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