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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Complex analysis
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Fatou Type Theorems - Maximal Functions and Approach Regions (Paperback, 1998 ed.)
Loot Price: R1,374
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Fatou Type Theorems - Maximal Functions and Approach Regions (Paperback, 1998 ed.)
Series: Progress in Mathematics, 147
Expected to ship within 18 - 22 working days
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A basic principle governing the boundary behaviour of holomorphic
func tions (and harmonic functions) is this: Under certain growth
conditions, for almost every point in the boundary of the domain,
these functions ad mit a boundary limit, if we approach the
bounda-ry 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 half spaces (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 ho)omor phic 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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