Some new multipliers of Fourier series

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 $b\upsilon = \left \{ {\lambda \left | {{\lambda _n} = {\lambda _{ - n}},\sum \nolimits _{k = 0}^\infty {\left | {\Delta {\lambda _k}} \right |} + {{\sup }_n}\left | {{\lambda _n}} \right | < \infty } \right .} \right \}$, where $\Delta {\lambda _k} = {\lambda _k} - {\lambda _{k + 1}}$ and the space of even bounded quasiconvex sequences is defined by $q = \left \{ {\lambda \left | {{\lambda _n} = {\lambda _{ - n}},\sum \nolimits _{k = 1}^\infty {k\left | {{\Delta ^2}{\lambda _k}} \right | + {{\sup }_n}\left | {{\lambda _n}} \right | < \infty } } \right .} \right \}$, 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.