Given any $ g \in L^\infty(\mathbb{R})$ , define the associated multiplier operator $ T_g \colon L^2(\mathbb{R}) \to L^2(\mathbb{R})$ by $ $ \mathcal{F}(T_g f) \ = \ g.\mathcal{F}f $ $ where $ \mathcal{F}$ denotes the Fourier transform.

Definition. We will say that a function $ f \in L^\infty(\mathbb{R})$ is nice if for every bounded $ g \in C^\infty(\mathbb{R})$ there exists a measurable function $ \tilde{T}_{\!g}f \in L^0(\mathbb{R})$ such that for every $ \phi \in C_c^\infty(\mathbb{R})$ and $ \tau \in \mathbb{R}$ with $ \phi=1$ on a neighbourhood of $ \tau$ , the function $ $ f_s \ := \ T_g(t \mapsto f(t)\phi(\tfrac{t-\tau}{s})) $ $ converges in probability to $ \tilde{T}_{\!g}f$ as $ s \to \infty$ .

(The “convergence in probability” here is defined with respect to any finite measure on $ \mathbb{R}$ that’s equivalent to the Lebesgue measure.)

Is there any straightforward characterisation of nice functions? Or at least, is there any important large class of functions (beyond absolutely integrable bounded functions) all of whose members are nice functions?