Abstract
Our starting point is a theorem of de Leeuw and Rudin that describes the extreme points of the unit ball in the Hardy space H 1 . We extend this result to subspaces of H 1 formed by functions with smaller spectra. More precisely, given a finite set 𝒦 of positive integers, we prove a Rudin–de Leeuw type theorem for the unit ball of H 𝒦 1 , the space of functions f∈H 1 whose Fourier coefficients f ^(k) vanish for all k∈𝒦.
Highlights
Introduction and main resultLet T stand for the unit circle {ζ ∈ C : |ζ| = 1}, endowed with normalized Lebesgue measure, and let L1 = L1(T) be the space of all complex-valued integrable functions f on T, with norm f 1 := 2π | f (ζ)| |d ζ|
Recall that a non-null function F ∈ H 1 is said to be outer if log |F (0)| =
A priori, one feels that the extreme points f of ball(HK1 ) should probably be the unit-norm functions which are fairly close to being outer, in some sense or other
Summary
A priori, one feels that the extreme points f of ball(HK1 ) should probably be the unit-norm functions which are fairly close to being outer, in some sense or other. It turns out that if f ∈ ball(HK1 ) is extreme, its inner factor, I , must be a finite Blaschke product whose degree (i.e., the number of its zeros) does not exceed M (= #K). This means that I is writable, possibly after multiplication by a unimodular constant, as m. There, we supplement Theorem 1 with a result concerning the exposed points of ball(HK1 )
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
