Skip to main content Accessibility help
×
Home
Hostname: page-component-cf9d5c678-xvx2z Total loading time: 0.326 Render date: 2021-07-29T16:59:07.716Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Asymptotic first boundary value problem for holomorphic functions of several complex variables

Published online by Cambridge University Press:  18 May 2021

Paul M. Gauthier
Affiliation:
Département de mathématiques et de statistique, Université de Montréal, Montréal, QC H3C 3J7, Canada
Mohammad Shirazi
Affiliation:
Department of Mathematics and Statistics, McGill University, Montréal, QC H3A 0B9, Canada e-mail: mohammad.shirazi@mail.mcgill.ca

Abstract

In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle ${\mathbb T}$ , there is a function f holomorphic in the unit disc ${{\mathbb D}}$ , having $\psi $ as radial limit a.e. on ${\mathbb T}$ . We consider an analogous boundary value problem, where the unit disc is replaced by a Stein domain on a complex manifold and radial approach to a boundary point p is replaced by (asymptotically) total approach to p.

Type
Article
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The present research was supported by NSERC (Canada) grant RGPIN-2016-04107.

References

Alexander, H. and Wermer, J., Several complex variables and Banach algebras. 3rd ed., Graduate Texts in Mathematics, 35, Springer, New York, 1998.Google Scholar
Anderson, B. D. O. and Vongpanitlerdm, S., Network analysis and synthesis: a modern systems theory approach. Prentice Hall, Englewood Cliffs, NJ, 1973.Google Scholar
Bayart, F. and Gauthier, P. M., Functions universal for all translation operators in several complex variables. Canad. Math. Bull. 60(2017), no. 3, 462469.CrossRefGoogle Scholar
Falcó, J. and Gauthier, P. M., An asymptotic holomorphic boundary problem on arbitrary open sets in Riemann surfaces. J. Approx. Theory 257(2020), 105451.CrossRefGoogle Scholar
Falcó, J. and Gauthier, P. M., Approximation in measure: Dirichlet problem, universality and the Riemann hypothesis. Izv. Ross. Akad. Nauk Ser. Mat. 85(2021), no. 3, 222238. Izv. Math. 85, no. 3, 1–15.Google Scholar
Falcó, J. and Gauthier, P. M., Asymptotic first boundary problem for elliptic operators. Submitted, 2021.Google Scholar
Izzo, A. J., Polynomial hulls of arcs and curves. Proc. Amer. Math. Soc. 149(2021), 199207.CrossRefGoogle Scholar
Kallin, E., Fat polynomially convex sets . 1966 Function algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965), pp. 149152 Scott-Foresman, Chicago, III.Google Scholar
Lehto, O., On the first boundary value problem for functions harmonic in the unit circle. Ann. Acad. Sci. Fenn. Ser. A. I. 1955(1955), no. 210, 26 pp.Google Scholar
Noshiro, K., Cluster sets, Ergebnisse der Mathematik und ihrer Grenzgebiete. N. F., Heft, 28, Springer, Berlin, Göttingen, and Heidelberg, 1960.Google Scholar
Rudin, W.. Real and complex analysis. 3rd ed. McGraw-Hill, New York, 1987.Google Scholar
Wage, M. L., A generalization of Lusin's theorem. Proc. Amer. Math. Soc. 52(1975), 327332.Google Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Asymptotic first boundary value problem for holomorphic functions of several complex variables
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Asymptotic first boundary value problem for holomorphic functions of several complex variables
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Asymptotic first boundary value problem for holomorphic functions of several complex variables
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *