The Poisson kernel for sl(3, ℝ)

AbstractLet P(r, θ) be the two-dimensional Poisson kernel in the unit disc D. It is proved that there exists a special sequence {ak} of points of D which is non-tangentially dense for ∂D and such that any function on ∂D can be expanded in series of P(|ak|, (·)–arg ak) with coefficients depending continuously on f in various classes of functions. The result is used to solve a Cauchy-type problem for Δu = μ, where μ is a measure supported on {ak}.

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Szegö Kernel and Poisson Kernel

Mathematics and Its Applications - Theory of Complex Homogeneous Bounded Domains ◽

10.1007/1-4020-2133-x_8 ◽

2007 ◽

pp. 329-360

Keyword(s):

Poisson Kernel ◽

Szegö Kernel ◽

Szegő Kernel ◽

Szego Kernel

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Vector Field Decompositions using Multiscale Poisson Kernel

IEEE Transactions on Visualization and Computer Graphics ◽

10.1109/tvcg.2020.2984413 ◽

2020 ◽

pp. 1-1

Author(s):

Harsh Bhatia ◽

Robert M. Kirby ◽

Valerio Pascucci ◽

Peer-Timo Bremer

Keyword(s):

Vector Field ◽

Poisson Kernel

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Some dual aspects of the Poisson kernel

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500018149 ◽

1990 ◽

Vol 33 (2) ◽

pp. 207-232 ◽

Cited By ~ 1

Author(s):

F. F. Bonsall

Keyword(s):

Continuous Function ◽

Borel Measure ◽

Poisson Kernel ◽

Infinite Subset ◽

Open Unit ◽

Positive Borel Measure ◽

Positive Continuous Function ◽

Open Unit Disc ◽

Counting Measure ◽

Countably Infinite

The Poisson kernel is defined for z in the open unit disc D and ζ in the unit circle ∂D. As usually employed, it is integrated with respect to the second variable and a measure on ∂D to yield a harmonic function on D. Here, we fix a σ-finite positive Borel measure m on D and integrate the Poisson kernel with respect to the first variable against a function φ in L1(m) to obtain a function Tmφ on ∂D. We ask for what measures m the range of Tm is L1(∂D), for what m the kernel of Tm is non-zero, and for what m every positive continuous function on ∂D is of the form Tmφ with φ non-negative. When m is the counting measure of a countably infinite subset {ak:k∈ℕ} of D, the function (Tmφ)(ζ) is of the form with . The main results generalize results previously obtained for sums of this form. A related mapping from Lp(m) into Lp(∂D) with 1 <p<∞ is briefly considered.

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