Compactness theorems of measurable selections and integral representation theorem

1977 ◽  
pp. 125-161
Author(s):  
Charles Castaing ◽  
Michel Valadier
Keyword(s):  
Integral Representation ◽  
Representation Theorem ◽  
Measurable Selections ◽  
Compactness Theorems

scholarly journals Linear operators on Köthe spaces of vector fields

2013 ◽  
Vol 21 (2) ◽  
pp. 53-80
Author(s):  
Ion Chiţescu ◽  
Liliana Sireţchi
Keyword(s):  
Integral Representation ◽  
Representation Theorem ◽  
Vector Fields ◽  
Linear Operators ◽  
Köthe Spaces

Abstract The study of Köthe spaces of vector fields was initiated by the present authors. In this paper linear operators on these spaces are studied. An integral representation theorem is given and special types of linear operators are introduced and studied.

On seismic deghosting using integral representation for the wave equation: Use of Green’s functions with Neumann or Dirichlet boundary conditions

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. T89-T98 ◽  
Author(s):  
Lasse Amundsen ◽  
Arthur B. Weglein ◽  
Arne Reitan
Keyword(s):  
Wave Equation ◽  
Integral Representation ◽  
Boundary Conditions ◽  
Representation Theorem ◽  
Green's Functions ◽  
Dirichlet Boundary ◽  
Green’S Functions ◽  
Dirichlet Boundary Conditions ◽  
Sea Surface ◽  
Receiver Side

The recent interest in broadband seismic technology has spurred research into new and improved seismic deghosting solutions. One starting point for deriving deghosting methods is the representation theorem, which is an integral representation for the wave equation. Recent research results show that by using Green’s functions with Dirichlet boundary conditions in the representation theorem, source-side deghosting of already receiver-side deghosted wavefields can be achieved. We found that the choice of Green’s functions with Neumann boundary conditions on the sea surface and the plane that contains the sources leads to an identical but simpler solution with fewer processing steps. In addition, we found that pressure data can be receiver-side deghosted by introducing Green’s functions with Dirichlet boundary conditions on the sea surface and the plane containing the receivers into a modified representation theorem. The deghosting methods derived from the representation theorem are wave-theoretic algorithms defined in the frequency-space domain and can accommodate streamers of any shape (e.g., slanted). Our theoretical analysis of deghosting is performed in the frequency-wavenumber domain where analytical deghosting solutions are well known and thus are available for verifying the solutions. A simple numerical example can be used to show how source-side deghosting can be performed in the space domain by convolving data with Green’s functions.

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