Fatou Type Theorems: Maximal Functions and Approach Regions
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Boston, MA
Birkhäuser Boston
1998
|
| Schriftenreihe: | Progress in Mathematics
147 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | A basic principle governing the boundary behaviour of holomorphic functions (and harmonic functions) is this: Under certain growth conditions, for almost every point in the boundary of the domain, these functions admit a boundary limit, if we approach the boundary point within certain approach regions. For example, for bounded harmonic functions in the open unit disc, the natural approach regions are nontangential triangles with one vertex in the boundary point, and entirely contained in the disc [Fat06]. In fact, these natural approach regions are optimal, in the sense that convergence will fail if we approach the boundary inside larger regions, having a higher order of contact with the boundary. The first theorem of this sort is due to J. E. Littlewood [Lit27], who proved that if we replace a nontangential region with the rotates of any fixed tangential curve, then convergence fails. In 1984, A. Nagel and E. M. Stein proved that in Euclidean halfspaces (and the unit disc) there are in effect regions of convergence that are not nontangential: These larger approach regions contain tangential sequences (as opposed to tangential curves). The phenomenon discovered by Nagel and Stein indicates that the boundary behaviour of holomorphic functions (and harmonic functions), in theorems of Fatou type, is regulated by a second principle, which predicts the existence of regions of convergence that are sequentially larger than the natural ones |
| Beschreibung: | 1 Online-Ressource (XII, 154 p) |
| ISBN: | 9781461223108 9781461274964 |
| DOI: | 10.1007/978-1-4612-2310-8 |
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| 490 | 1 | |a Progress in Mathematics |v 147 | |
| 500 | |a A basic principle governing the boundary behaviour of holomorphic functions (and harmonic functions) is this: Under certain growth conditions, for almost every point in the boundary of the domain, these functions admit a boundary limit, if we approach the boundary point within certain approach regions. For example, for bounded harmonic functions in the open unit disc, the natural approach regions are nontangential triangles with one vertex in the boundary point, and entirely contained in the disc [Fat06]. In fact, these natural approach regions are optimal, in the sense that convergence will fail if we approach the boundary inside larger regions, having a higher order of contact with the boundary. The first theorem of this sort is due to J. E. Littlewood [Lit27], who proved that if we replace a nontangential region with the rotates of any fixed tangential curve, then convergence fails. In 1984, A. Nagel and E. M. Stein proved that in Euclidean halfspaces (and the unit disc) there are in effect regions of convergence that are not nontangential: These larger approach regions contain tangential sequences (as opposed to tangential curves). The phenomenon discovered by Nagel and Stein indicates that the boundary behaviour of holomorphic functions (and harmonic functions), in theorems of Fatou type, is regulated by a second principle, which predicts the existence of regions of convergence that are sequentially larger than the natural ones | ||
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Datensatz im Suchindex
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| ctrlnum | (OCoLC)879624694 (DE-599)BVBBV042420012 |
| dewey-full | 515.9 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
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| dewey-search | 515.9 |
| dewey-sort | 3515.9 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-1-4612-2310-8 |
| format | Electronic eBook |
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| id | DE-604.BV042420012 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:06Z |
| institution | BVB |
| isbn | 9781461223108 9781461274964 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027855429 |
| oclc_num | 879624694 |
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| physical | 1 Online-Ressource (XII, 154 p) |
| psigel | ZDB-2-SMA ZDB-2-BAE ZDB-2-SMA_Archive |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Birkhäuser Boston |
| record_format | marc |
| series | Progress in Mathematics |
| series2 | Progress in Mathematics |
| spelling | Biase, Fausto Verfasser aut Fatou Type Theorems Maximal Functions and Approach Regions by Fausto Biase Boston, MA Birkhäuser Boston 1998 1 Online-Ressource (XII, 154 p) txt rdacontent c rdamedia cr rdacarrier Progress in Mathematics 147 A basic principle governing the boundary behaviour of holomorphic functions (and harmonic functions) is this: Under certain growth conditions, for almost every point in the boundary of the domain, these functions admit a boundary limit, if we approach the boundary point within certain approach regions. For example, for bounded harmonic functions in the open unit disc, the natural approach regions are nontangential triangles with one vertex in the boundary point, and entirely contained in the disc [Fat06]. In fact, these natural approach regions are optimal, in the sense that convergence will fail if we approach the boundary inside larger regions, having a higher order of contact with the boundary. The first theorem of this sort is due to J. E. Littlewood [Lit27], who proved that if we replace a nontangential region with the rotates of any fixed tangential curve, then convergence fails. In 1984, A. Nagel and E. M. Stein proved that in Euclidean halfspaces (and the unit disc) there are in effect regions of convergence that are not nontangential: These larger approach regions contain tangential sequences (as opposed to tangential curves). The phenomenon discovered by Nagel and Stein indicates that the boundary behaviour of holomorphic functions (and harmonic functions), in theorems of Fatou type, is regulated by a second principle, which predicts the existence of regions of convergence that are sequentially larger than the natural ones Mathematics Global analysis (Mathematics) Functions of complex variables Differential equations, partial Functions of a Complex Variable Analysis Several Complex Variables and Analytic Spaces Mathematik Progress in Mathematics 147 (DE-604)BV000004120 147 https://doi.org/10.1007/978-1-4612-2310-8 Verlag Volltext |
| spellingShingle | Biase, Fausto Fatou Type Theorems Maximal Functions and Approach Regions Progress in Mathematics Mathematics Global analysis (Mathematics) Functions of complex variables Differential equations, partial Functions of a Complex Variable Analysis Several Complex Variables and Analytic Spaces Mathematik |
| title | Fatou Type Theorems Maximal Functions and Approach Regions |
| title_auth | Fatou Type Theorems Maximal Functions and Approach Regions |
| title_exact_search | Fatou Type Theorems Maximal Functions and Approach Regions |
| title_full | Fatou Type Theorems Maximal Functions and Approach Regions by Fausto Biase |
| title_fullStr | Fatou Type Theorems Maximal Functions and Approach Regions by Fausto Biase |
| title_full_unstemmed | Fatou Type Theorems Maximal Functions and Approach Regions by Fausto Biase |
| title_short | Fatou Type Theorems |
| title_sort | fatou type theorems maximal functions and approach regions |
| title_sub | Maximal Functions and Approach Regions |
| topic | Mathematics Global analysis (Mathematics) Functions of complex variables Differential equations, partial Functions of a Complex Variable Analysis Several Complex Variables and Analytic Spaces Mathematik |
| topic_facet | Mathematics Global analysis (Mathematics) Functions of complex variables Differential equations, partial Functions of a Complex Variable Analysis Several Complex Variables and Analytic Spaces Mathematik |
| url | https://doi.org/10.1007/978-1-4612-2310-8 |
| volume_link | (DE-604)BV000004120 |
| work_keys_str_mv | AT biasefausto fatoutypetheoremsmaximalfunctionsandapproachregions |