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Some New Multipliers of Fourier Series

Martin Buntinas
Proceedings of the American Mathematical Society
Vol. 101, No. 3 (Nov., 1987), pp. 497-502
DOI: 10.2307/2046396
Stable URL: http://www.jstor.org/stable/2046396
Page Count: 6
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Some New Multipliers of Fourier Series
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Abstract

Let L1 be the space of all complex-valued 2π-periodic integrable functions f and let $\widehat{L^1}$ be the space of sequences of Fourier coefficients f̂. A sequence λ is an $(\widehat{L^1} \rightarrow \widehat{L^1})$ multiplier if λ · f̂ = (λ(n)f̂(n)) belongs to $\widehat{L^1}$ for every f in L1. The space of even sequences of bounded variation is defined by $bv = \{ \lambda\mid\lambda_n = \lambda_{-n}, \sum^\infty_{k = 0}|\Delta\lambda_k| + \sup_n|\lambda_n| < \infty \}$, where Δλk = λk - λk + 1 and the space of even bounded quasiconvex sequences is defined by $q = \{\lambda\mid\lambda_n = \lambda_{-n}, \sum^\infty_{k = 1} k|\Delta^2\lambda_k| + \sup_n|\lambda_n| < \infty \}$, where Δ2λk = Δλk - Δλk + 1. It is well known that $q \subset bv$ and $q \subset (\widehat{L^1} \rightarrow \widehat{L^1})$ but $bv \not\subset (\widehat{L^1} \rightarrow \widehat{L^1})$. This result is significantly improved by finding an increasing family of sequence spaces dvp between q and bv which are $(\widehat{L^1} \rightarrow \widehat{L^1})$ multipliers. Since the $(\widehat{L^1} \rightarrow \widehat{L^1})$ multipliers are the 2π-periodic measures, this result gives sufficient conditions for a sequence to be the Fourier coefficients of a measure.

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