Carleson's convergence theorem for Dirichlet series
(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)
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
- 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
- 2022-04-22 18:48:36
@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) < +&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.}}, 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}}, }