Статья «BINOMIALS WHOSE DILATIONS GENERATE H2(D), "Алгебра и анализ"»

This note is about the completeness of the function families {z{\-z) ·. n= 1,2,...} in the Hardy space Яо(Ю), and some related questions. It is shown that for |A| > R(N) the family is complete in Hq(D) (and often is a Riesz basis of Hq), whereas for |A| < r(N) it is not, where both radii r(N) < R(N) tends to infinity and behave more or less as N (as N -> то). Several results are also obtained for more general binomials {z" (1 - jz") : n = 1,2,... } where |A| ^ 1 and v G C. The author is indebted to Alexander Borichev for pointing out a few bugs in an initial version of the manuscript, and to Boris Mityagin for a discussion stimulated the author's interest to the question.