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Carleson's convergence theorem for Dirichlet series

Hedenmalm, Håkan LU and Saksman, Eero LU (2003) In Pacific Journal of Mathematics 208(1). p.85-109
Abstract
A Hilbert space of Dirichlet series is obtained by considering the Dirichlet series f(s) = Sigma(n=1)(infinity) a(n)n(-s) that satisfy Sigma(n=0)(infinity) a(n)(2) < +&INFIN;. These series converge in the half plane Re s > 1/2 and define a functions that are locally L-2 on the boundary Re s > 1/2. An analog of Carleson's celebrated convergence theorem is obtained: Each such Dirichlet series converges almost everywhere on the critical line Re s = 1/2. To each Dirichlet series of the above type corresponds a trigonometric series Sigma(n=1)(infinity) a(n)chi(n), where chi is a multiplicative character from the positive integers to the unit circle. The space of characters is naturally identified with the infinite-dimensional torus... (More)
A Hilbert space of Dirichlet series is obtained by considering the Dirichlet series f(s) = Sigma(n=1)(infinity) a(n)n(-s) that satisfy Sigma(n=0)(infinity) a(n)(2) < +&INFIN;. These series converge in the half plane Re s > 1/2 and define a functions that are locally L-2 on the boundary Re s > 1/2. An analog of Carleson's celebrated convergence theorem is obtained: Each such Dirichlet series converges almost everywhere on the critical line Re s = 1/2. To each Dirichlet series of the above type corresponds a trigonometric series Sigma(n=1)(infinity) a(n)chi(n), where chi is a multiplicative character from the positive integers to the unit circle. The space of characters is naturally identified with the infinite-dimensional torus T-infinity, where each dimension comes from a a prime number. The second analog of Carleson's theorem reads: The above trigonometric series converges for almost all characters chi. (Less)
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author
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Contribution to journal
publication status
published
subject
in
Pacific Journal of Mathematics
volume
208
issue
1
pages
85 - 109
publisher
Pacific Journal of Mathematics
external identifiers
  • wos:000180227100007
  • scopus:0037289537
ISSN
0030-8730
language
English
LU publication?
yes
id
e8365cf9-c9a5-422f-9fea-9cd581ba0208 (old id 319722)
alternative location
http://pjm.math.berkeley.edu/pjm/2003/208-1/p07.xhtml
date added to LUP
2016-04-01 15:58:58
date last changed
2021-08-04 01:49:31
@article{e8365cf9-c9a5-422f-9fea-9cd581ba0208,
  abstract     = {A Hilbert space of Dirichlet series is obtained by considering the Dirichlet series f(s) = Sigma(n=1)(infinity) a(n)n(-s) that satisfy Sigma(n=0)(infinity) a(n)(2) &lt; +&amp;INFIN;. These series converge in the half plane Re s &gt; 1/2 and define a functions that are locally L-2 on the boundary Re s &gt; 1/2. An analog of Carleson's celebrated convergence theorem is obtained: Each such Dirichlet series converges almost everywhere on the critical line Re s = 1/2. To each Dirichlet series of the above type corresponds a trigonometric series Sigma(n=1)(infinity) a(n)chi(n), where chi is a multiplicative character from the positive integers to the unit circle. The space of characters is naturally identified with the infinite-dimensional torus T-infinity, where each dimension comes from a a prime number. The second analog of Carleson's theorem reads: The above trigonometric series converges for almost all characters chi.},
  author       = {Hedenmalm, Håkan and Saksman, Eero},
  issn         = {0030-8730},
  language     = {eng},
  number       = {1},
  pages        = {85--109},
  publisher    = {Pacific Journal of Mathematics},
  series       = {Pacific Journal of Mathematics},
  title        = {Carleson's convergence theorem for Dirichlet series},
  url          = {http://pjm.math.berkeley.edu/pjm/2003/208-1/p07.xhtml},
  volume       = {208},
  year         = {2003},
}