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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

##### MSC:
 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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