On fourier transforms of measures with compact support

Beurling, Arne; Malliavin, Paul

doi:10.1007/bf02545792

“…Density concepts were used by Paley and Wiener [PW34] and by Levinson [Lev40] to study spectral gaps and the uniqueness of nonharmonic Fourier series. Beurling's definition of density was used in the beautiful characterization of complete sets of exponentials due to Beurling and Malliavin [BM62], [BM67], and by Landau in his studies of completeness properties of exponentials [Lan64], and the sampling and interpolation of entire functions [Lan67].…”

Section: Irregular Gabor Systemsmentioning

confidence: 99%

Differentiation and the Balian-Low Theorem

Benedetto

¹

,

Heil

²

,

Walnut

³

1994

J Fourier Anal Appl

View full text Add to dashboard Cite

“…Now we turn to Theorem BM2 which, apparently, was the main motivation behind the work described in [BM1].…”

Section: Some Connections and Applications Of Theorem Bm1mentioning

confidence: 99%

“…In [BM1], this compound question was answered exhaustively: W p does not depend on p and coincides with the set W of all measurable weights ω, 0 < ω ≤ 1, for which L(ω) > −∞, and…”

Section: 4mentioning

confidence: 99%

Beurling–Malliavin multiplier theorem: The seventh proof

Mashreghi

¹

,

Nazarov

²

,

Havin

³

2006

St. Petersburg Math. J.

View full text Add to dashboard Cite

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) =

show abstract

“…Its depth and importance will be discussed thoroughly in Subsections 0.5, 1.1, 1.3, and 3.2. Now we only note that, since its first appearance in [BM1] in 1962, this theorem (apparently, designed for the needs of the celebrated Second Theorem BM2 about the completeness of a system of harmonics) has been a subject of permanent attention of analysts.…”

Section: 3mentioning

confidence: 99%

“…Yet, definitely, at least six proofs of Theorem BM1 had been known before, and the texts treating this result in various ways run to hundreds of pages (see [BM1,Koo1,Koo2,Koo3,Koo5,DeB,HJ,Mal] and also the reference lists in [Koo2,Koo3]). …”

Section: 3mentioning

confidence: 99%

Untitled

2006

St. Petersburg Math. J.

1

0

View full text Add to dashboard Cite

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) =

show abstract

On fourier transforms of measures with compact support

Cited by 159 publications

References 2 publications

Differentiation and the Balian-Low Theorem

Differentiation and the Balian-Low Theorem

Beurling–Malliavin multiplier theorem: The seventh proof

Untitled

Contact Info

Product

Resources

About