Beurling–Malliavin multiplier theorem: The seventh proof

Abstract

Abstract. We present a new proof of the Beurling-Malliavin theorem, often called the "multiplier theorem", concerning the existence of a real-valued function on R with spectrum in a given (small) interval and with a given small majorant of the modulus. This proof pertains entirely to real analysis. It only involves elementary facts about the Hilbert transformation; neither complex variable methods nor potential theory is exploited. The heart of the proof is Theorem 2, which treats preservation of the Lipschitz property under the Hilbert transformation. We also include a short survey of earlier proofs of the Beurling-Malliavin theorem and its generalizations to model (coinvariant) subspaces of the Hardy space H 2 (R)."...Yet the seventh proof of that exists, reliable beyond doubt! And it will be shown you in a while." [Bu, p. 453] §0. Introduction 0.1. Our purpose in this (mainly expository) paper is to present a new, relatively simple and self-contained proof of the Beurling-Malliavin multiplier theorem. Sometimes it is called "The first Beurling-Malliavin theorem", and we shall use the symbol BM1 to refer to it. Theorem BM2, lying beyond the scope of this paper, will be touched upon in Subsection 3. (X.Y means Subsection Y of §X).We shall reduce Theorem BM1 to a certain problem about the Hilbert transformation on the real line. The only new result in the present paper is Theorem 2 in Subsection 2.1, proved by F. L. Nazarov. The reduction of Theorem BM1 to that statement is contained in [HMII] and, implicitly,in [Koo2, p. 502]. Here we reproduce this reduction along the lines of [HMII], but with great simplifications. In particular, our presentation is free of the so-called model subspaces of the Hardy space H 2 (R), which were the main topic of [HMI] and [HMII] (we shall talk about them separately in Subsection 3.5). We wanted to give a focused and generally accessible proof of Theorem BM1, avoiding anywhere near important references to other sources. Our paper is written for nonexperts. We only assume the knowledge of elementary properties of the Fourier transformation on R. For "good" functions f , the Fourier transformation is defined by the formulâ f (ξ) =

On the multidimensional Nazarov lemma

“…This theorem, sometimes called Nazarov's lemma was crucial in [6] for the beautiful "geodesic" proof of the so-called First Beurling-Malliavin theorem. The latter theorem is among the deepest results in harmonic analysis.…”

“…The latter theorem is among the deepest results in harmonic analysis. To illustrate this, we mention the articles [6], [7], [5], [2], [3], [1], where Theorem A was shown to apply many results in the theory of exponential systems. We especially emphasize that Theorem A is a cornerstone of the proof of the famous Second Beurling-Malliavin Theorem.…”

“…Remark 3. Our proof of Theorem 1 relies on the one dimensional pattern of [6]. However, there is a subtlety: the construction of the regularized system of intervals from [6] which is the crucial step of the proof of Theorem A should be nontrivially adapted for the case of several dimensions (see Remark 6 and Theorem 3).…”

“…Our proof of Theorem 1 relies on the one dimensional pattern of [6]. However, there is a subtlety: the construction of the regularized system of intervals from [6] which is the crucial step of the proof of Theorem A should be nontrivially adapted for the case of several dimensions (see Remark 6 and Theorem 3). Moreover, we show that the regularization is a quite general operation, since it can be successfully applied to an arbitrary family of mutually disjoint dyadic squares, see Remark 5.…”

Bump Functions With Monotone Fourier Transforms Satisfying Decay Bounds

“…The Beurling-Malliavin multiplier theorem (see e.g. [1]) guarantees the existence of smooth, compactly supported functions whose Fourier transforms have sub-exponential decay (i.e. an exponential of a power less than one).…”