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Il Nuovo Cimento (1955-1965)

, Volume 30, Issue 6, pp 1452–1464 | Cite as

Analyticity and phase retrieval

  • P. Roman
  • A. S. Marathay
Article

Summary

The general problem of constructing a spectrumg(v) from the knowledge of the magnitude of its Fourier transform ¦ γr) ¦ is considered. The question reduces to locating the zeros of the analytic continuation γ(τ) in the upper half-plane. It is shown that ifg(v) is real, the complex zeros of γ(τ) in the u.h.p. either are on the imaginary axis or occur pairwise in a symmetrical position. If, in addition, g(v)≥0, the zeros on the imaginary axis disappear. The conditiong(v) ≥ 0 also leads to the requirement that γ(τ) must be representable as a convolution of a functionh(τ) with itself. The analytic properties ofh(τ) and the equations to determine it are discussed. Possible ways to obtain the solution of the ensuing nonlinear eigenvalue problem are suggested.

Riassunto

Si tratta il problema della costruzione di uno spettrog(v) conoscendo la grandezza della sua trasformata di Fourier ¦ γτqg)¦. La questione si riduce ad individuare la posizione degli zeri della contiuuazione analitica γ(τ) nel semipiano superiore. Si dimostra che seg(v) è reale, gli zeri complessi di γ(τ) nel semipiano superiore o sono sull’asse immaginario o si riscontrano a coppie in posizioni simmetriche. Se inoltreg(v) ≥ 0, gli zeri sull’asse immaginario scompaiono. La condizioneg(v)≥0 comporta anche la necessità che γ(τ) sia rappresentabile come una convoluzione di una funzioneh(τ) con se stessa. Si discutono le proprietà analitiche dih(τ) e le equazioni per determinarla. Si suggeriscono alcuni possibili metodi per ottenere la soluzione.

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References

  1. (1).
    See, for example,E. C. Titchmaksh:Theory of Fourier Integrals (Oxford, 1937), p. 127–128 and p. 125; orJ. S. Toll:Phys. Rev.,104, 1760 (1956); orJ. Hilgevoobd:Dispersion Belations and Causal Description (Amsterdam, 1960), p. 34–36.Google Scholar
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Copyright information

© Società Italiana di Fisica 1963

Authors and Affiliations

  • P. Roman
    • 1
  • A. S. Marathay
    • 1
  1. 1.Department of PhysicsBoston UniversityBoston

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