Excesses of systems of exponential functions

A. M. Sedletskii

Izvestiya. Mathematics , Tome 16 (1981) no. 1, pp. 191-205.

Résumé

A nonnegative sequence $\{\alpha_n\}$ is called an admissible majorant if the condition $|\lambda_n-\mu_n|\leqslant\alpha_n$, where $\{\lambda_n\}$ and $\{\mu_n\}$ are real regular sequences, implies that the systems of functions $\{\exp(i\lambda_nx)\}$ and $\{\exp(i\mu_nx)\}$ have the same excess in the space $L^2(-a,a)$ ($a\infty$). A complete characterization of admissible majorants is given for the class of sequences $\alpha_n\downarrow0$ that have the smoothness property $\alpha_{n+1}\sim\alpha_n$. This is used to establish the definitiveness of the author's criterion for the stability of the excess of a system of exponentials in $L^2$. Bibliography: 10 titles.

Bibliographie

[1] Sedletskii A. M., “Izbytki sistem pokazatelnykh funktsii”, Matem. zametki, 22:6 (1977), 803–814

[2] Alexander W. O., Redheffer R., “The excess of sets of complex exponentials”, Duke Math. J., 34:1 (1967), 59–72 | DOI | MR | Zbl

[3] Levinson N., Gap and density theorems, Amer. Math. Soc., N.,Y., 1940 | MR | Zbl

[4] Zigmund A., Trigonometricheskie ryady, t. I, Mir, M., 1965 | MR

[5] Polya G., “Über die Nullstellen gewisser ganzer Funktionen”, Math. Z., 2 (1918), 352–383 | DOI | MR | Zbl

[6] Boas R. P., Entire functions, Academic press, N.,Y., 1954 | MR | Zbl

[7] Sedletskii A. M., “O raznosti izbytkov dvukh sistem pokazatelnykh funktsii”, Trudy MEI, 357 (1978), 98–102 | MR

[8] Goldman M. L., “Obobschennye yadra drobnogo poryadka”, Differentsialnye uravneniya, 7:12 (1971), 2199–2210 | MR

[9] Peterson D., “The excess of sets of complex exponentials”, Proc. Amer. Math. Soc., 44:2 (1974), 321–325 | DOI | MR | Zbl

[10] Elsner J., “Zulässige Abänderungen von Exponentialsistemen im $L^p(-A,A)$”, Math. Z., 120:3 (1971), 211–220 | DOI | MR | Zbl