Conditions for making a function with Fourier transform to an absolutely integrable function

Question

There are many non absolutely integrable functions with convergent Fourier transform, i.e., $$X(f) = \int_{-\infty}^{+\infty}x(t)\exp(-2\pi ift)dt \ \ \text{converges but } \int_{-\infty}^{+\infty}|x(t)|dt \ \ \text{diverges.}$$ For example take $x(t) = \text{sinc}(t)$ or $x(t) = \sin(t^2)$. I'm looking for minimal conditions which makes a Fourier transformable function to an absolutely integrable function. For example what happens if we know $X(f)$ is rational or $X(f)$ decays to $0$ as $f \to +\infty$? I think this problem is important from engineering point of view since an LTI system is BIBO-stable if and only if its impulse response is absolutely integrable, so it's useful to know equivalent conditions.

Answer 1

There is indeed a significant asymmetry between the conditions guaranteeing the existence of a Fourier transform and those ensuring that the original function is absolutely integrable in $L^1(\mathbb{R})$. The Fourier transform of an $L^1(\mathbb{R})$ function is bounded, uniformly continuous, and vanishes at infinity by the Riemann–Lebesgue lemma, but none of these properties suffice in reverse.

As you note, functions such as $\text{sinc}(t) = \frac{\sin(\pi t)}{\pi t}$ and $\sin(t^2)$ provide examples. The former lies in $L^2(\mathbb{R}) \setminus L^1(\mathbb{R})$, yet its Fourier transform is the rectangle function, compactly supported and hence rapidly decaying. The latter, though neither in $L^1(\mathbb{R})$ nor $L^2(\mathbb{R})$, has a Fourier transform that exists in the distributional sense. These cases highlight the need for more nuanced sufficient conditions under which the Fourier transform data $X(f)$ implies $x(t) \in L^1(\mathbb{R})$.

Here are three analytically robust sufficient conditions for absolute integrability of $x(t) = \mathcal{F}^{-1}\{X\}$:

1. Smoothness Condition (Sobolev): If $X$ belongs to the Sobolev space $W^{2,1}(\mathbb{R})$, i.e., $X \in L^1(\mathbb{R})$ and its second derivative $X^{\prime\prime} \in L^1(\mathbb{R})$, then $x(t) \in L^1(\mathbb{R})$. This follows by integrating by parts twice in the inversion formula; the boundary terms vanish due to the integrability and decay properties implied by $X, X^{\prime\prime} \in L^1(\mathbb{R})$, and we obtain $$x(t) = \frac{-1}{4\pi^2 t^2} \mathcal{F}^{-1}\{X^{\prime\prime}\}(t),$$ with the inverse transform of $X^{\prime\prime}$ being bounded (since $X^{\prime\prime} \in L^1(\mathbb{R})$). Hence $|x(t)| = O(t^{-2})$ as $|t| \to \infty$, which guarantees integrability for the continuous function $x(t)$.

2. Decay Condition: Suppose $X(f)$ is sufficiently smooth and $$|X(f)| \le \frac{C}{(1 + |f|)^{\alpha}}, \quad \text{for some } C>0, \alpha > 1, \text{ and large } |f|.$$ Then $X \in L^1(\mathbb{R})$. Under mild regularity assumptions, its derivatives decay faster; in particular, $|X^{\prime\prime}(f)| \lesssim |f|^{-\alpha-2}$, ensuring $X^{\prime\prime} \in L^1(\mathbb{R})$ since $\alpha+2 > 3 > 1$. Thus, the smoothness condition applies. This decay threshold $\alpha > 1$ is sharp.

3. Rationality Condition: If $X(f) = P(f)/Q(f)$ is a rational function with no poles on the real axis and $\deg(Q) \ge \deg(P)+2$, then $x(t) \in L^1(\mathbb{R})$. The pole condition ensures $X(f)$ is smooth and bounded on $\mathbb{R}$, and the relative degree condition ensures that $|X(f)| = O(|f|^{-k})$ with $k \ge 2$, satisfying the decay condition. The inverse Fourier transform can then be computed via residues and gives a linear combination of exponentially decaying functions.

Each of these conditions guarantees a sufficiently strong combination of decay and regularity in $X(f)$ to force integrability of $x(t)$. Notably, none of these are necessary conditions—there is no simple characterization of the Wiener algebra $A(\mathbb{R}) = \mathcal{F}(L^1(\mathbb{R}))$ in terms of classical pointwise properties of $X(f)$. The image $A(\mathbb{R})$ is a proper subset of $C_0(\mathbb{R})$ (continuous functions vanishing at infinity).

From an engineering perspective, these results directly inform BIBO stability criteria for continuous-time linear time-invariant (LTI) systems. An LTI system is BIBO-stable if and only if its impulse response $h(t) \in L^1(\mathbb{R})$. If the system’s transfer function $H(f)$ is rational, the rationality condition above translates directly to verifiable conditions on pole locations and relative degree.