1964-15 - ON SOME APPLICATIONS OF PROBABILITY TO ANALYSIS...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ON SOME APPLICATIONS OF PROBABILITY TO ANALYSIS AND NUMBER THEORY P . ERDS It would be quite impossible to give a survey of these subjects in a short article or lecture, and I will only succeed by making some arbitrary restrictions on the topics with which I will deal . First of all, I will restrict myself to problems and results on which I worked, and secondly, I will not discuss subjects which have been discussed in recently appeared review articles [1]. Probabilistic methods have been used in analysis for several decades ; it suffices to name Paley, Wiener, Kolmogoroff, Zygmund, Salem, Steinhaus, Kac, Dvoretzky, Kahane, and many others. I will restrict myself to some questions my collaborators and I worked on for several ar years. Hardy was the first to give an example of a power series E a,,, z " k k=1 cA which converges uniformly in z a < 1 but for which ak = oo. Piranian k=1 asked me for what sequences of integers n1< n2 < ... does there exist a power series ak zn k which converges uniformly in I z < 1 but for which k=1 z Z I ak = 00 . I proved [2] by probabilistic methods that if the sequence k=1 {n1 .} satisfies lim inf (n; - n 1) 1 i(i- 0 = 1 where j-i-->oo then if Y a,. z111-- converges for k--1 z (1) then such a power series exists. Zygmund [3] proved that ifnk+1/nk> c > 1 1, Z I a, < oo . Thus (1) is certainly k=1 not far from being best possible, and it is quite likely that it is, in fact, best possible ; in fact, Zygmund's theorem may remain true for every sequence which does not satisfy (1), in other words, for every sequence {nk}for which there exists an absolute constant c so that for every i < j n;-nti> (1+c)'- ti. (2) Curiously enough, (1) occurred in a seemingly different context . Gaier and Meyer-Knig [4] call the radius defined by z = re 2 O, 0 <it <1, U) singular for f (z) = Z ak zn if f (z) is unbounded in every sector I z I < 1, 71=1 Co ~- E < arg z <0-f- E where E > 0. They showed that if f (z) _ a,, znk k=1 and nk+1/n k > c > 1, and if f (z) is unbounded in (z ~ < 1, then every radius Received 1 May, 1963 . [JOURNAL LONDON MATH . Soc ., 39 (1964), 692-696] APPLICATIONS OF PROBABILITY TO ANALYSIS AND NUMBER THEORY 693 is a singular radius. Rnyi and I [5] showed by probabilistic methods W that if {n,;}satisfies (1) then there exists a power seriesf (z)= E ak znk k=1 m for which a k > 0, ak = oo, thus the positive real axis is a singular k=7 radius but no other radius is singular. In fact, f (z) isbounded in I z < 1 if a region I z-1 < e is excluded (for every e > 0). It again seems quite possible that our theorem is best possible;in fact, perhaps if{n k }satisfies (2) then the theorem of Gaier and Meyer-Kng remains true, but we could prove nothing in this direction.could prove nothing in this direction....
View Full Document

This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell University (Engineering School).

Page1 / 5

1964-15 - ON SOME APPLICATIONS OF PROBABILITY TO ANALYSIS...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online