How far can a function be “simplified” without altering its essential properties? In harmonic analysis, this question translates into understanding what happens when its Fourier transform is truncated outside a given domain. A new study extends a celebrated classical result and shows that, even in contexts as different as Lie groups and matrices, the geometry of the boundary of that domain remains the key factor.

Fourier analysis is one of the fundamental tools of modern mathematics and of numerous scientific and technological applications, from signal processing to quantum physics. It makes it possible to decompose a function into its basic frequencies and study its behaviour through them. But what happens when part of that information is removed and the Fourier transform is truncated outside a certain domain?

Since the 1970s, it has been known that this can drastically alter the properties of a function. A classical result by Charles Fefferman, published in 1971, showed that truncating the Fourier transform to a region of Euclidean space preserves a highly significant property of the function, namely its (L^p)-integrability, provided that the boundary of the domain is piecewise flat. This theorem established a surprising link between the geometry of a set and the analytical behaviour of the associated functions.

However, many questions remained open. What does this notion of “flatness” mean when the space is not Euclidean, but rather a curved Lie group? And what happens if, instead of functions, one truncates the entries of a matrix? How does the shape of the domain then influence the singular values of the resulting matrix?

A new study answers these seemingly disparate questions through a unifying approach. In their work, Javier Parcet (ICMAT-CSIC), Mikael de la Salle (CNRS, Université de Lyon) and Eduardo Tablate (KU Leuven) provide a complete characterisation of the local geometry of all truncations that preserve (L^p)-integrability in two settings that generalise Euclidean harmonic analysis: Fourier multipliers on Lie groups and Schur multipliers on matrices — matrix transformations defined by entrywise multiplication.

The central result shows that, in the matrix setting, the key lies in a surprisingly flexible notion of boundary flatness — much broader than those considered previously — from which Fefferman’s classical theorem and its generalisations to arbitrary Lie groups can be derived.

In the matrix case, all admissible truncations turn out to be differentiable variations of a well-known fundamental example: the triangular projection, also known as the matrix Hilbert transform. For Lie groups, there are three basic models: the classical Hilbert transform and two new Hilbert transforms defined on affine groups and universal covering groups. These examples serve as elementary building blocks from which all possible cases can be explained.

The work generalises Fefferman’s theorem to non-trigonometric and non-commutative settings and, in the case of Lie groups, completes a long-standing programme of research. It also reveals a deep connection between Schur multipliers and Euclidean Fourier multipliers, showing that they share a common structure even in the absence of an underlying Fourier transform.

Reference

Parcet J, de la Salle M, Tablate E. The local geometry of idempotent Schur multipliers. Forum of Mathematics, Pi. 2025;13:e14. doi:10.1017/fmp.2025.6