Beurling–Malliavin multiplier theorem

Statement

Let $\omega:\mathbb R\to\mathbb R_{\le0}$ be Lipschitz and suppose

\int_{\mathbb R}\frac{\omega(x)}{1+x^2}\,dx>-\infty.

For every $\sigma>0$ , there is a nonzero $f\in L^2(\mathbb R)$ such that

\operatorname{supp}\widehat f\subseteq[-\sigma,\sigma], \qquad |f(x)|\le e^{\omega(x)}\quad(x\in\mathbb R).

Meaning

The weight prescribes where and how rapidly the multiplier should decay. Lipschitz regularity rules out arbitrarily sharp oscillation, while the logarithmic integral is the sharp global growth condition. The conclusion balances this decay against bounded Fourier support.

Complex-analytic viewpoint

By the Paley–Wiener theorem, the problem is to construct an entire function of controlled exponential type under the majorant $e^\omega$ on the real axis. Classical proofs use subharmonic majorants, outer functions, or carefully placed zeros.

References

Arne Beurling and Paul Malliavin, “On Fourier transforms of measures with compact support,” Acta Mathematica 107 (1962), 291–309. DOI record.
Javad Mashreghi, Fedor Nazarov, and Victor Havin, “The Beurling–Malliavin multiplier theorem: the seventh proof,” St. Petersburg Mathematical Journal 17 (2006), 699–744. DOI record.