Let \(m\) be a measurable bounded function, \(S\) a bounded functional and \(m(\xi) S(\xi)^{it-1}\) a Fourier multiplier on \(L^p\) uniformly in \(t\in\mathbb{R}\). It is shown that \(m\log(S)^k\) is a Fourier multiplier on \(L^p\) for every positive integer \(k\). This assertion is proved with the aid of the other main result: Theorem. Let \(T\) be a linear operator such that, for every \(\theta\), there exists a bounded Calderón family of operators \(\overline{L^\theta}: (L^{p_0}, L^{p_1})\to (L^{q_0}, L^{q_1})\) so that \(\theta^n(L^\theta)^{(n)}_\theta f\) converges to \(Tf\) almost everywhere as \(\theta\) tends to zero, for every \(f\in L^{p_0}\cap L^{p_1}\). Then \(T: L^{p_0}\to L^{q_0}\) is bounded. The latter result is derived via a detailed investigation of the behaviour of the Schechter interpolation spaces as \(\theta\) tends to zero.