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\begin{document}$$X({\mathbb {R}})$$\end{document} be a separable rearrangement-invariant space and w be a suitable Muckenhoupt weight. We show that for any semi-almost periodic Fourier multiplier a on X(R,w)={f:fw∈X(R)}\documentclass[12pt]{minimal}
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\begin{document}$$X({\mathbb {R}},w)=\{f:fw\in X({\mathbb {R}})\}$$\end{document} there exist uniquely determined almost periodic Fourier multipliers al,ar\documentclass[12pt]{minimal}
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\begin{document}$$a_l,a_r$$\end{document} on X(R,w)\documentclass[12pt]{minimal}
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\begin{document}$$X({\mathbb {R}},w)$$\end{document}, such that a=(1-u)al+uar+a0,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} a=(1-u)a_l+ua_r+a_0, \end{aligned}$$\end{document}for some monotonically increasing function u with u(-∞)=0\documentclass[12pt]{minimal}
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\begin{document}$$u(-\infty )=0$$\end{document}, u(+∞)=1\documentclass[12pt]{minimal}
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\begin{document}$$u(+\infty )=1$$\end{document} and some continuous and vanishing at infinity Fourier multiplier a0\documentclass[12pt]{minimal}
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\begin{document}$$a_0$$\end{document} on X(R,w)\documentclass[12pt]{minimal}
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\begin{document}$$X({\mathbb {R}},w)$$\end{document}. This result extends previous results by Sarason (Duke Math J 44:357–364, 1977) for L2(R)\documentclass[12pt]{minimal}
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\begin{document}$$L^2({\mathbb {R}})$$\end{document} and by Karlovich and Loreto Hernández (Integral Equ Oper Theor 62:85–128, 2008) for weighted Lebesgue spaces Lp(R,w)\documentclass[12pt]{minimal}
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\begin{document}$$L^p({\mathbb {R}},w)$$\end{document} with weights in a suitable subclass of the Muckenhoupt class Ap(R)\documentclass[12pt]{minimal}
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\begin{document}$$A_p({\mathbb {R}})$$\end{document}.
