Analytic solutions of convolution equations on convex sets in the complex plane with an open obstacle on the boundary

2020 ◽  
Vol 211 (7) ◽  
pp. 1014-1040
Author(s):  
S. N. Melikhov ◽  
L. V. Khanina
Keyword(s):  
Complex Plane ◽  
Convex Sets ◽  
Analytic Solutions ◽  
Convolution Equations

scholarly journals Convex-Cyclic Weighted Composition Operators and Their Adjoints

2021 ◽  
Vol 18 (4) ◽  
Author(s):  
Tesfa Mengestie
Keyword(s):  
Complex Plane ◽  
Pointwise Convergence ◽  
Convex Sets ◽  
Fock Space ◽  
Composition Operators ◽  
Invariant Subspaces ◽  
Weighted Composition Operators ◽  
Weighted Composition ◽  
Convergence Topology

AbstractWe characterize the convex-cyclic weighted composition operators $$W_{(u,\psi )}$$ W ( u , ψ ) and their adjoints on the Fock space in terms of the derivative powers of $$ \psi $$ ψ and the location of the eigenvalues of the operators on the complex plane. Such a description is also equivalent to identifying the operators or their adjoints for which their invariant closed convex sets are all invariant subspaces. We further show that the space supports no supercyclic weighted composition operators with respect to the pointwise convergence topology and, hence, with the weak and strong topologies, and answers a question raised by T. Carrol and C. Gilmore in [5].

Solutions of Convolution Equations in Convex Sets

1987 ◽  
Vol 109 (3) ◽  
pp. 521 ◽  
Author(s):  
Carlos A. Berenstein ◽  
Daniele C. Struppa
Keyword(s):  
Convex Sets ◽  
Convolution Equations

scholarly journals On continuation of quasi-analytic solutions of partial differential equations to compact convex sets

1985 ◽  
Vol 39 (3) ◽  
pp. 306-316 ◽  
Author(s):  
Wojciech Abramczuk
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Convex Sets ◽  
Analytic Solution ◽  
Compact Convex ◽  
Analytic Solutions ◽  
Scalar Case ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Constant Coefficients

AbstractIn the early 70s A. Kaneko studied the problem of continuation of regular solutions of systems of linear partial differential equations with constant coefficients to compact convex sets. We show here that the conditions be obtained for real analytic solutions also hold in the quasi-analytic case. In particular we show that every quasi-analytic solution of the system p(D)u = 0 defined outside a compact convex subset K or Rn can be continued as a quasi-analytic solution to K if and only if the system is determined and the -module Ext1(Coker p′, ) has no elliptic component; here is the ring of polynomials in n variables, p is a matrix with elements from and p′ is the transposed matrix. In the scalar case, i.e. when p is a single polynomial, these conditions mean that p has no elliptic factor.

On the Use of POCS for Restoring the “Missing Wedge” in Electron Tomography

1990 ◽  
Vol 48 (1) ◽  
pp. 520-521
Author(s):  
Neng-Yu Zhang ◽  
Bruce F. McEwen ◽  
Joachim Frank
Keyword(s):  
Function Space ◽  
Convex Sets ◽  
Electron Tomography ◽  
Spinal Ganglion ◽  
Fourier Space ◽  
Missing Information ◽  
Voltage Electron ◽  
High Voltage Electron Microscope ◽  
Energy Bound ◽  
Spatial Boundedness

Reconstructions of asymmetric objects computed by electron tomography are distorted due to the absence of information, usually in an angular range from 60 to 90°, which produces a “missing wedge” in Fourier space. These distortions often interfere with the interpretation of results and thus limit biological ultrastructural information which can be obtained. We have attempted to use the Method of Projections Onto Convex Sets (POCS) for restoring the missing information. In POCS, use is made of the fact that known constraints such as positivity, spatial boundedness or an upper energy bound define convex sets in function space. Enforcement of such constraints takes place by iterating a sequence of function-space projections, starting from the original reconstruction, onto the convex sets, until a function in the intersection of all sets is found. First applications of this technique in the field of electron microscopy have been promising.To test POCS on experimental data, we have artificially reduced the range of an existing projection set of a selectively stained Golgi apparatus from ±60° to ±50°, and computed the reconstruction from the reduced set (51 projections). The specimen was prepared from a bull frog spinal ganglion as described by Lindsey and Ellisman and imaged in the high-voltage electron microscope.

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