Summary
The general problem of constructing a spectrumg(v) from the knowledge of the magnitude of its Fourier transform ¦ γ(τ r) ¦ 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
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The research reported in this paper was sponsored by the U. S. Air Force Office of Scientific Eesearch, under Contract no. AF-AFOSR-248.
This author was financially supported by the U. S. Army Eesearch Office (Durham), under Contract no. DA-ARO(D)-31-124-Gr260.
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Roman, P., Marathay, A.S. Analyticity and phase retrieval. Nuovo Cim 30, 1452–1464 (1963). https://doi.org/10.1007/BF02749823
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DOI: https://doi.org/10.1007/BF02749823