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On pure imaginary perturbations of the exponents \(\lambda _ n\) in the system \(\{\exp (i\lambda _ nt)\}\). (Russian) Zbl 0582.42010
If \(\Lambda =\{\lambda_ n\}\) is a sequence of complex numbers we denote by \(E_ 2(\Lambda)\) the surplus in \(L^ 2=L^ 2(-\pi,\pi)\) of exponential functions in the system \(\{\exp (i\lambda_ nt)\}\) defined in the following way: if the given system is complete (incomplete) in \(L^ 2\) \(E_ 2(\Lambda)\) is the number (taken with the minus sign) of exponents by whose elimination (adjunction) the new system becomes complete and minimal. In a previous paper J. Elsner [Math. Z. 120, 211-220 (1971; Zbl 0197.387)] proved: If \(\Lambda =\{\lambda_ n\}\) and \(M=\{\mu_ n\}\) are in the strip \(\{z=x+iy:\) \(| y| \leq h<\infty \}\) and (*) \(Re \lambda_ n=Re \mu_ n\) for each n, then \(E_ 2(\Lambda)=E_ 2(M).\)
In the present paper the author proves: (i) Let \(\phi (x)>0\) on [0,\(\infty)\) and \(\phi\) (x)\(\uparrow \infty\) as \(x\to \infty\). Then there exist sequences \(\Lambda\) and M in \(\{\) \(z: | y| \leq \phi (| x|)\}\) such that (*) holds and \(E_ 2(\Lambda)\neq E_ 2()\). (ii) Let \(\Lambda\) and M be in \(\{\) \(z: | y| \leq \phi (| x|)\}\) where \(\phi\) is positive and nondecreasing on [0,\(\infty)\) and \(\sum^{\infty}_{n=1}\phi^ 2(n)/n^ 2<\infty\). If (*) holds then \(| Im \lambda_ n| \geq | Im \mu_ n|\) implies \(E_ 2(\Lambda)\geq E_ m(M)\).
Reviewer: S.Aljančić

42A65 Completeness of sets of functions in one variable harmonic analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
30B60 Completeness problems, closure of a system of functions of one complex variable
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