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Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 637–658 | Cite as

On the Completeness of Gaussians in a Hilbert Functional Space

  • Victor KatsnelsonEmail author
Article
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Abstract

Let \(w_{T}(t)\) and \(w_{\Omega }(\omega )\) be non-negative functions defined for \(t\in \mathbb {R}\) and \(\omega \in \mathbb {R}\), where \(\mathbb {R}=(-\infty ,\infty )\). Some regularity conditions are posed on these functions. The space \(\mathcal {H}_{w_T,w_{\Omega }}\) consists of those functions \(x\) for which \(\int \nolimits _{-\infty }^{\infty }|x(t)|^2w_{T}(t)dt+ \int \nolimits _{-\infty }^{\infty }|\hat{x}(\omega )|^2w_{\Omega }(\omega )d\omega = \Vert x\Vert ^2_{\mathcal {H}_{w_T,w_{\Omega }}}<\infty \), where \(\hat{x}\) is the Fourier transform of the function \(x\). We show that the system of Gaussians \(\big \lbrace \exp (-\alpha (t-\tau )^2)\big \rbrace \), where \(\alpha \) runs over \(\mathbb {R}^{+}=(0,+\infty )\) and \(\tau \) runs over \(\mathbb {R}\), is a complete system in the space \(\mathcal {H}_{w_T,w_{\Omega }}\).

Keywords

Fourier transform Hilbert functional space Sobolev space Gaussians Time-frequency analysis 

List of symbols

\(\mathbb {R}\)

The set of all real numbers.

\(\mathbb {R}_{+}\)

The set of all strictly positive real numbers.

\(\mathbb {C}\)

The set of all complex numbers.

\(d\xi ,\,dt,\,d\omega \)

The Lebesgue measure on the real axis \(\mathbb {R}\) (if the variable on \(\mathbb {R}\) is denoted by \(\xi ,\,t,\,\omega \) respectively).

Mathematics Subject Classification

Primary 46E20 46E35 Secondary 41A28 42B10 

References

  1. 1.
    Bogachev, V.I.: Measure Theory, vol. I. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
  2. 2.
    Dym, H., McKean, H.P.: Fourier Series and Integrals. Academic Press, New York (1972)zbMATHGoogle Scholar
  3. 3.
    Gröchenig, K.: Weight functions in time-frequency analysis. In: Pseudo-Differential Operators: Partial Differential Equations and Time–Frequency Analysis, pp. 343–366. Fields Institute Communications Volume 52, American Mathematical Society, Providence, RI (2007)Google Scholar
  4. 4.
    Halmos, P.R.: Measure Theory. Springer, New York (1974)zbMATHGoogle Scholar
  5. 5.
    Halmos, P.R., Sunder, V.S.: Bounded Integral Operators on $L^2$ Spaces. Springer, Berlin (1978)CrossRefzbMATHGoogle Scholar
  6. 6.
    Nazarov, F.L.: Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type. St. Petersburg Math. J. 5(4), 663–717 (1994)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe Weizmann InstituteRehovotIsrael

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