Functions with given estimate for $\partial f/\partial\overline z$, and N.~Levinson's theorem

E. M. Dyn'kin

Sbornik. Mathematics, Tome 18 (1972) no. 2, pp. 181-189

Résumé

In this paper it is shown that a twice continuously differentiable function $\varphi$ on the unit circle with Fourier coefficients $\{\widehat\varphi(n)\}$ admits a continuously differentiable extension $f$ to the whole plane such that $$ \frac{\partial f}{\partial\overline z}=O[h(|1-|z||)] $$ (here $h$ is a given weight with $h(+0)=0)$ if $\varphi(n)=O(n^{-1}a_n)$, where $$ a_n=\int_0^1h(r)(1-r)^{|n|}\,dr,\qquad n=0,\pm1,\pm2,\dots\,. $$ If $\int_0\ln\ln\frac1{h(r)}\,dr+\infty$, then the class of such functions $\varphi$ turns out to be non-quasi-analytic. Hence a new proof of the known theorem of N. Levinson on the normality of families of analytic functions is derived. Bibliography: 7 titles.