Luzin examples

in the theory of functions of a complex variable

Examples that characterize boundary uniqueness properties of analytic functions (see [1], [2]).

1) For any set $ E $ of measure zero on the unit circle $ \Gamma = \{ {z } : {| z | = 1 } \} $, N.N. Luzin constructed (1919, see [1]) a function $ f ( z) $ that is regular, analytic and bounded in the unit disc $ D = \{ {z } : {| z | < 1 } \} $ and is such that $ f ( z) $ does not have radial boundary values along each of the radii that end at points of $ E $.

A similar example of Luzin and I.I. Privalov (1925, see [2], [3]) differs only by insignificant changes.

2) Luzin also constructed (1925, see [2]) regular analytic functions $ f _ {1} ( z) $ and $ f _ {2} ( z) \not\equiv 0 $ in $ D $ that tend, respectively, to infinity and zero along all radii that end at points of some set of full measure $ 2 \pi $ on $ \Gamma $. This set $ E $ is of the first Baire category (cf. Baire classes) on $ \Gamma $.

See also Boundary properties of analytic functions; Luzin–Privalov theorems; Cluster set.

References

[1]	N.N. Luzin, , Collected works , 1 , Moscow (1953) pp. 267–269 (In Russian)
[2]	N.N. Luzin, , Collected works , 1 , Moscow (1953) pp. 280–318 (In Russian)
[3]	I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[4]	A. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauki i Tekhn. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian)

References

[a1]	E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9