Wronskians of Fourier and Laplace transforms. (English) Zbl 1476.11117
C. Andréief’s identity for Gram determinants [Bord. Mém. (3) II. 1–14 (1886; JFM 18.0262.03)] is used to give an expression of the Wronskian of the first \(n\) derivatives of a Fourier or Laplace transform. Wiener’s \(L^1\) tauberian theorem is then used to give a necessary and sufficient criterion for a Fourier transform \(\varphi\) of some special type to have only real zeros. This restatement is in terms of the density in \(L^1(\mathbb R)\) of subspaces of translates of functions related to \(\varphi\). In particular, an equivalent form of the Riemann hypothesis is given.
Reviewer: Michel Balazard (Marseille)
MSC:
| 11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 30D10 | Representations of entire functions of one complex variable by series and integrals |
| 30D15 | Special classes of entire functions of one complex variable and growth estimates |
| 40E05 | Tauberian theorems |
| 42A82 | Positive definite functions in one variable harmonic analysis |
Keywords:
Fourier transform; Laplace transform; Wronskian; entire function; Laguerre-Pólya class; Riemann hypothesisCitations:
JFM 18.0262.03References:
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