Journal of Soviet Mathematics

, Volume 16, Issue 3, pp 1118–1139 | Cite as

Lectures on the shift operator. IV

  • N. K. Nikol'skii


This is the continuation of the papers I-III published in Vols. 39, 47, and 56 of the same “Zapiski.” One considers the shift operator f → zf in the spaces of (generalized) functions on the circumference T and their spectral subspaces. If X is such a space and e⊂T, then
. What is the class of all X-negligible sets e, i.e., such that Xe={O} ? This is the wellknown question in harmonic analysis regarding sets of uniqueness. We prove known theorems about such sets (due to O. Frostman, D. Newman, and Y. Katznelson) as well as new ones. Among these: if
$$X = \left\{ {f.\sum\limits_{n \in \mathbb{Z}} {\left| {\hat f(n)} \right|^p } \log ^{ - \beta p} (\left| n \right| + 1)< \infty } \right\}, 0 = \beta< 1 \leqslant p< \frac{2}{{1 + \beta }}$$
, then there exists a set e with Xe={O} for which the Lebesgue measure of Te is arbitrarily small.


Harmonic Analysis Lebesgue Measure Shift Operator Spectral Subspace 
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© Plenum Publishing Corporation 1981

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  • N. K. Nikol'skii

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