Fourier Transform Inequality

Question

I am trying to find conditions that the real-valued function $f$ should satisfy in order to guarantee an inequality of the form: $$ \left| \mathcal{F} f (\lambda) \right| \leq |g(\lambda)|. $$ Here $\mathcal{F}f$ denotes the Fourier transform of $f$: $$ \mathcal{F}f(\lambda) = \int e^{- i \lambda x} f(x) dx. $$

Answer 1

Look at the Paley-Wiener theorem: $f$ is a (complex valued) $L^2$-function with $\mathtt{supp}f=\left[-A,A\right]$

if and only if for $\bar{f}\left(\xi\right)\;:=\;\intop_{-\infty}^{\infty}f\left(x\right)\cdot e^{ix \xi}\mathtt{d}x,\;\xi\in\mathbb{C}$ the following hold:

i) $\left|\bar{f}\right|$ is an entire function.

ii) $\exists C\;\forall\xi\in\mathbb{C}\;:\;\left|\bar{f}\left(\xi\right)\right|\leq e^{A\cdot\left|\xi\right|}$ ("$\bar{f}$ is of exponential type $A$").

iii) $\forall y\in\mathbb{R}\;:\;\intop_{-\infty}^{\infty}\left|\bar{f}\left(x+i\cdot y\right)\right|^{2}\mathtt{d}x<\infty$

Note that the growth constant $A$ yields a sharp estimate for the support and vice versa: Among the functions $f$ with the property that the magnitude of the (inverse) FT $\bar{f}$ is a function of exponential type $A$ there are always some with a support that spans the whole interval $\mathtt{supp}f=\left[-A,A\right]$ and conversely if $\mathtt{supp}f=\left[-A,A\right]$ the inverse FT will be of exponential type $A$.

There is a multitude of Paley-Wiener theorems, they form a whole class of theorems relating the growth of a complex function to the support of its fourier transform. Google is your friend here, good luck finding one that fits your needs!

EDIT:

For real $f$ also this simple relationship may help (you probably already know it): $$\left|\hat{f}\left(\omega\right)\right|^{2}=\hat{f}\left(\omega\right)\cdot\overline{\hat{f}\left(\omega\right)}\underset{f\mathrm{\; real}}{=}\hat{f}\left(\omega\right)\cdot\hat{f}\left(-\omega\right)=\mathcal{F}\left(\left(f\left(\cdot\right)*f\left(-\cdot\right)\right)\left(t\right)\right)=\mathcal{F}\left(\intop_{-\infty}^{\infty}f\left(\xi\right)*f\left(\xi-t\right)\mathtt{d}\xi\right) $$ Assuming the integral exists. -- The keywords here are "autocorrelation" and "energy spectral density".

Also note Plancharel's theorem ($f$ may be complex-valued again): $$\intop_{-\infty}^{\infty}\left|\hat{f}\left(\omega\right)\right|^{2}\mathtt{d}\omega=\intop_{-\infty}^{\infty}\left|f\left(t\right)\right|^{2}\mathtt{d}t $$

Assuming the integrals exist.

Finally, here is a question of mine with respect to Paley-Wiener. This question (and my own answer) contain some additional formulae, that you may look at - maybe it is useful...: Designing a FIR filter in frequency domain: Paley-Wiener "the other way around".