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An uncertainty principle for the windowed Bochner–Fourier transform with the complex-valued window function

  • Mykola Ivanovich Yaremenko EMAIL logo
Published/Copyright: February 20, 2024
Journal of Applied Analysis
From the journal Journal of Applied Analysis Volume 30 Issue 2

Abstract

The generalization of Hardy uncertainty principle for the windowed Bochner–Fourier transform on the Heisenberg group is proved. We consider a Bochner measurable function Ψ : G → X , where X is a completely separable Hilbert space X and Let G be a completely separable, unimodular, connected nilpotent Lie group. We establish that if ϕ ∈ C C ⁢ ( G ) is a non-trivial window function and Ψ ∈ L 2 ⁢ ( G ) satisfies ∥ V ϕ ⁢ ( Ψ ) ⁢ ( g , χ ) ∥ HB ≤ c 2 ⁢ ( g ) ⁢ exp ⁡ ( - β ⁢ ∥ χ ∥ 2 ) , β > 0 , then Ψ = 0 almost everywhere.

Keywords: Windowed Fourier transform; uncertainty principle; Heisenberg inequality; locally compact group; Bochner integral; Bochner space; Hardy group
MSC 2020: 28B05; 28B10; 46B25; 46B50

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Received: 2023-10-06
Revised: 2023-12-24
Accepted: 2024-01-09
Published Online: 2024-02-20
Published in Print: 2024-12-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On the Laplace transform
  3. An uncertainty principle for the windowed Bochner–Fourier transform with the complex-valued window function
  4. Controllability result in α-norm for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces
  5. Polynomial convergence of iterations of certain random operators in Hilbert space
  6. Trigonometric approximation of signals (functions) belonging to certain Lipschitz spaces using C δ.C operator
  7. Almost compactness in neutrosophic topological spaces
  8. Some results for a weakly coupled system of semi-linear structurally damped σ-evolution equations
  9. On the stochastic elliptic equations involving fractional derivative
  10. A note on Köthe–Toeplitz duals and multiplier spaces of sequence spaces involving bicomplex numbers
  11. A brief survey on the development and applications of Goebel’s coincidence point theorem in differential and integral equations
  12. Boundary controllability for variable coefficients one-dimensional wave equation with interior degeneracy
  13. On random pairwise comparisons matrices and their geometry
  14. Statistically convergent difference sequences of bi-complex numbers
  15. On some inequalities concerning polynomials with restricted zeros
  16. New retarded nonlinear integral inequalities of Gronwall–Bellman–Pachpatte type and their applications
  17. On a new variant of cyclic (noncyclic) condensing operators with existence of optimal solutions to an FDE
  18. Exponential stability of non-conformable fractional-order systems
https://doi.org/10.1515/jaa-2023-0120
Keywords for this article
Windowed Fourier transform; uncertainty principle; Heisenberg inequality; locally compact group; Bochner integral; Bochner space; Hardy group

Articles in the same Issue

  1. Frontmatter
  2. On the Laplace transform
  3. An uncertainty principle for the windowed Bochner–Fourier transform with the complex-valued window function
  4. Controllability result in α-norm for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces
  5. Polynomial convergence of iterations of certain random operators in Hilbert space
  6. Trigonometric approximation of signals (functions) belonging to certain Lipschitz spaces using C δ.C operator
  7. Almost compactness in neutrosophic topological spaces
  8. Some results for a weakly coupled system of semi-linear structurally damped σ-evolution equations
  9. On the stochastic elliptic equations involving fractional derivative
  10. A note on Köthe–Toeplitz duals and multiplier spaces of sequence spaces involving bicomplex numbers
  11. A brief survey on the development and applications of Goebel’s coincidence point theorem in differential and integral equations
  12. Boundary controllability for variable coefficients one-dimensional wave equation with interior degeneracy
  13. On random pairwise comparisons matrices and their geometry
  14. Statistically convergent difference sequences of bi-complex numbers
  15. On some inequalities concerning polynomials with restricted zeros
  16. New retarded nonlinear integral inequalities of Gronwall–Bellman–Pachpatte type and their applications
  17. On a new variant of cyclic (noncyclic) condensing operators with existence of optimal solutions to an FDE
  18. Exponential stability of non-conformable fractional-order systems
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