Let \(\mathcal {M}_{X(\mathbb {R})}\) be the Banach algebra of all Fourier multipliers on a Banach function space \(X(\mathbb {R})\) such that the Hardy–Littlewood maximal operator is bounded on \(X(\mathbb {R})\) and on its associate space \(X'(\mathbb {R})\). For two sets \(\varPsi ,\varOmega \subset \mathcal {M}_{X(\mathbb {R})}\), let \(\varPsi _\varOmega\) be the set of those \(c\in \varPsi\) for which there exists \(d\in \varOmega\) such that the multiplier norm of \(\chi _{\mathbb {R}\setminus [-N,N]}(c-d)\) tends to zero as \(N\rightarrow \infty\). In this case, we say that the Fourier multiplier c is equivalent at infinity to the Fourier multiplier d. We show that if \(\varOmega\) is a unital Banach subalgebra of \(\mathcal {M}_{X(\mathbb {R})}\) consisting of nice Fourier multipliers (for instance, continuous or slowly oscillating in certain sense) and \(\varPsi\) is an arbitrary unital Banach subalgebra of \(\mathcal {M}_{X(\mathbb {R})}\), then \(\varPsi _\varOmega\) is a also a unital Banach subalgebra of \(\mathcal {M}_{X(\mathbb {R})}\).

令\（\ mathcal {M} _ {X（\ mathbb {R}）} \）是Banach函数空间\（X（\ mathbb {R}）\）上所有傅立叶乘法器的Banach代数，使得Hardy –Littlewood最大运算符在\（X（\ mathbb {R}）\）及其关联空间\（X'（\ mathbb {R}）\）上有界。对于两个集合\（\ varPsi，\ varOmega \ subset \ mathcal {M} _ {X（\ mathbb {R}）} \），令\（\ varPsi _ \ varOmega \）是这些\（c \ \ varPsi \）中存在\（d \ in \ varOmega \）中，使得\（\ chi _ {\ mathbb {R} \ setminus [-N，N]}（cd）\）的乘数范数趋于设为\（N \ rightarrow \ infty \）为零。在这种情况下，我们说傅立叶乘数c在无限远处等于傅立叶乘数d。我们证明，如果\（\ varOmega \）是\（\ mathcal {M} _ {X（\ mathbb {R}）} \）的单位Banach子代数，则由良好的傅立叶乘法器组成（例如，连续或缓慢振荡）从某种意义上说），\（\ varPsi \）是\（\ mathcal {M} _ {X（\ mathbb {R}）} \）的任意单位Banach子代数，则\（\ varPsi _ \ varOmega \）是一个也是\（\ mathcal {M} _ {X（\ mathbb {R}）} \\的单位Banach子代数。