Carleson's convergence theorem for Dirichlet series
(2003) In Pacific Journal of Mathematics 208(1). p.85109 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 L2 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 infinitedimensional 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 L2 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 infinitedimensional torus Tinfinity, 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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/319722
 author
 Hedenmalm, Håkan ^{LU} and Saksman, Eero ^{LU}
 organization
 publishing date
 2003
 type
 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
 00308730
 language
 English
 LU publication?
 yes
 id
 e8365cf9c9a5422f9fea9cd581ba0208 (old id 319722)
 alternative location
 http://pjm.math.berkeley.edu/pjm/2003/2081/p07.xhtml
 date added to LUP
 20160401 15:58:58
 date last changed
 20210804 01:49:31
@article{e8365cf9c9a5422f9fea9cd581ba0208, 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 L2 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 infinitedimensional torus Tinfinity, 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 = {00308730}, language = {eng}, number = {1}, pages = {85109}, 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/2003/2081/p07.xhtml}, volume = {208}, year = {2003}, }