摘要:

Given a simply connected domain ω in the complex plane with boundary of Minkowski dimension α, we show that the area with respect to the Poincaré metric divided by the area with respect to the quasihyperbolic metric of the super level sets of the distance function to ∂ω is asymptotically bounded by α2as the distance decreases to zero. These bounds are established using a symmetrization argument, showing that the Poincaré area of a subdomain of ω can be estimated in terms of a conformal modulus of its complement. We find a generalization of this result to locally simply connected domains and also show that the curvature of the quasi-hyperbolic metric tends to -α on the average, as we approach ∂ω. This behavior of the curvature leads us to conjecture that the bound α2can be replaced by α.

ISSN:0278-1077    

DOI:10.1080/17476939308814690

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

Signal processing methods are placed in a function theoretic setting. The Dirichlet problem associated with the elliptic partial differential equationis an entire function, is considered on a bounded star-shaped domain ω ⊃Rnwith Lyapunov boundary. Solutions are constructed by sampling boundary data that are “band limited” and “finite energy” in the frequency regime. Lagrange type interpolation formulae follow.

ISSN:0278-1077    

DOI:10.1080/17476939308814691

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

Letfbe analytic in an unbounded open setDand assume that the cluster setCffat the boundary ∂Dis contained either in a closed setKwhich is convex at infinity or in a compact setK: in both casesC\Kis connected. We prove that if there existsw0∈f(D)\Ksuch thatf-1(w0) contains at leastppoints, then the maximum modulusM(r,f) = sup|z=|f(z)|zεD, must grow at a certain rate at infinity: ifKis unbounded, then; ifKis compact, thenHere c denotes constants and the inequalities hold for large values ofr.

ISSN:0278-1077    

DOI:10.1080/17476939308814692

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

ISSN:0278-1077    

DOI:10.1080/17476939308814679

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor