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Existence of the Bedrosian identity for Fourier multiplier operators

  • Rongrong Lin and Haizhang Zhang EMAIL logo
Published/Copyright: July 17, 2015
Forum Mathematicum
From the journal Forum Mathematicum Volume 28 Issue 4

Abstract

The Hilbert transform H satisfies the Bedrosian identity H(fg) =fHg whenever the supports of the Fourier transforms of f,g∈L2(ℝ) are respectively contained in A= [-a,b] and B=ℝ∖(-b,a), where 0≤a,b≤ +∞. Attracted by this interesting result arising from the time-frequency analysis, we investigate the existence of such an identity for a general bounded Fourier multiplier operator on L2(ℝd) and for general support sets A and B. A geometric characterization of the support sets for the existence of the Bedrosian identity is established. Moreover, the support sets for the partial Hilbert transforms are all found. In particular, for the Hilbert transform to satisfy the Bedrosian identity, the support sets must be given as above.

Keywords: Bedrosian identity; Fourier multiplier operators; translation-invariant operators; Hilberttransform; Riesz transforms
MSC 2010: 32A55

Funding statement: Supported in part by Natural Science Foundation of China under grants 11222103 and 11101438, and by US Army Research Office under grant W911NF-12-1-0163.

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Received: 2014-8-25
Revised: 2015-4-23
Published Online: 2015-7-17
Published in Print: 2016-7-1

© 2016 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Residues of standard intertwining operators on p-adic classical groups
  3. Anomalies on codimension growth of algebras
  4. Brackets with (τ,σ)-derivations and (p,q)-deformations of Witt and Virasoro algebras
  5. Bounded gaps between primes with a given primitive root, II
  6. Minimal potential results for the Schrödinger equation in a slab
  7. The classification of p-nilpotent restricted Lie algebras of dimension at most 4
  8. Global gradient estimates for nonlinear obstacle problems with nonstandard growth
  9. Existence of the Bedrosian identity for Fourier multiplier operators
  10. Projective bundles over small covers and the bundle triviality problem
  11. Gromov (non-)hyperbolicity of certain domains in ℂN
  12. The action of a compact Lie group on nilpotent Lie algebras of type {{n,2}}
  13. Restricted enveloping algebras whose skew and symmetric elements are Lie metabelian
https://doi.org/10.1515/forum-2014-0158
Keywords for this article
Bedrosian identity; Fourier multiplier operators; translation-invariant operators; Hilberttransform; Riesz transforms

Articles in the same Issue

  1. Frontmatter
  2. Residues of standard intertwining operators on p-adic classical groups
  3. Anomalies on codimension growth of algebras
  4. Brackets with (τ,σ)-derivations and (p,q)-deformations of Witt and Virasoro algebras
  5. Bounded gaps between primes with a given primitive root, II
  6. Minimal potential results for the Schrödinger equation in a slab
  7. The classification of p-nilpotent restricted Lie algebras of dimension at most 4
  8. Global gradient estimates for nonlinear obstacle problems with nonstandard growth
  9. Existence of the Bedrosian identity for Fourier multiplier operators
  10. Projective bundles over small covers and the bundle triviality problem
  11. Gromov (non-)hyperbolicity of certain domains in ℂN
  12. The action of a compact Lie group on nilpotent Lie algebras of type {{n,2}}
  13. Restricted enveloping algebras whose skew and symmetric elements are Lie metabelian
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