Given any $g \in L^\infty(\mathbb{R})$, we 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.

*I am interested in knowing how far beyond $L^2(\mathbb{R})$ it is possible to take a function $f \colon \mathbb{R} \to \mathbb{C}$, and it still make at least some kind of sense to talk about a corresponding function $T_gf \colon \mathbb{R} \to \mathbb{C}$. Accordingly, I have tried to formulate the weakest definition of the "domain $\mathfrak{D}(g)$ of definition of $T_g$" that I can, within the constraint that I still wish to be in the world of functions, not distributions*:

A sequence $\mathbf{r}\!=\!(r^{[k]})_{k \geq 1}$ of positive real numbers will be called a **valid growth rate** if there exists a test function $\phi \in C_c^\infty(\mathbb{R})$ such that $\phi(0)=\|\phi\|_\infty=1$ and for all $k \geq 1$, $\,\phi^{(k)}(0)=0$ and $\|\phi^{(k)}\|_\infty \leq r^{[k]}$.

Definition.For any $g \in L^\infty(\mathbb{R})$, let $\mathfrak{D}(g)$ be the set of all functions $f \in L_{\mathrm{loc}}^2(\mathbb{R})$ for which there exists a measurable function $\tilde{T}_{\!g}f \colon \mathbb{R} \to \mathbb{C}$, an increasing sequence of compact intervals $K_n \subset \mathbb{R}$ covering $\mathbb{R}$ and a sequence of valid growth rates $\mathbf{r}_n=(r_n^{[k]})$, such that the following holds: for every sequence of test functions $\phi_n \in C_c^\infty(\mathbb{R})$ with the properties that

- $\phi_n=1$ on $K_n$ for each $n$,
- $\|\phi_n\|_\infty=1$ and $\|\phi_n^{(k)}\|_\infty \leq r_n^{[k]}$ for each $n$ and $k$,
the sequence of functions $T_g(f\phi_n)$ converges in probability to $\tilde{T}_{\!g}f$ as $n \to \infty$.

The "convergence in probability" here is defined with respect to any probability measure on $\mathbb{R}$ that is equivalent to the Lebesgue measure. Alternatively, "convergence in probability" means that every subsequence admits a further subsequence converging Lebesgue-a.e. to the desired limit.

**Remark.** Formulating the definition in terms of convergence in probability (rather than something stronger like $L_{\mathrm{loc}}^p$-convergence) may be a bit risky, e.g. a sequence of zero-centred Gaussian PDFs whose standard deviation tends to zero converges in probability to the constant zero function.

Are there any existing results [or can we come up with some results] on how broad this class $\mathfrak{D}(g)$ is, for any particular class of functions $g$?

$\hspace{6mm}$For example: Is it the case that if $g$ is in some sense "sufficiently nice" (e.g. $g$ is smooth and each derivative has at most polynomial growth), then $\mathfrak{D}(g)$ includes the set of all bounded smooth functions all of whose derivatives are bounded?

**Physical motivation:** In signal processing, filters are typically multiplier operators, for which the multiplier $g$ is a rational function $P/Q$ where $\mathrm{order}(P) \leq \mathrm{order}(Q)$ and $Q$ has no real roots. Thus, in a sense, this question relates to the physical meaningfulness of filtering a signal which was recorded as just an extract from some more long-term process.

First thought: If $g$ is smooth and each derivative has at most polynomial growth, then $T_gf$ is a well-defined linear functional on $\mathcal{S}(\mathbb{R})$ for every $f \in L^\infty(\mathbb{R})$, by $$ T_gf(\bar{\varphi}) \ := \ \int_{\mathbb{R}} f(t)\overline{T_{\overline{g}}\varphi}(t) \, dt. $$ (I'm not sure if this fulfils the continuity requirement for being a tempered distribution.) In this case, I guess the question of whether $f \in \mathfrak{D}(g)$ can probably be reduced to whether Lebesgue-almost the whole real line can be covered by open sets $U$ for which $T_gf$ coincides with an $L_{\mathrm{loc}}^1$ function on $C_c^\infty(U)$.