A Generalized and Unified Framework of Local Fourier Analysis Using Matrix-Stencils

2021 ◽  
Vol 42 (3) ◽  
pp. 1096-1118
Author(s):  
Yunhui He
Keyword(s):  
Fourier Analysis ◽  
Unified Framework ◽  
Local Fourier Analysis

scholarly journals A unified mode decomposition method for physical fields in homogeneous cosmology

2014 ◽  
Vol 26 (03) ◽  
pp. 1430001 ◽  
Author(s):  
Zhirayr G. Avetisyan
Keyword(s):  
Fourier Transform ◽  
Field Equation ◽  
Scalar Field ◽  
Vector Bundle ◽  
Fourier Analysis ◽  
Homogeneous Spaces ◽  
Unified Framework ◽  
Mode Decomposition ◽  
Physical Fields ◽  
Homogeneous Vector Bundle

The methods of mode decomposition and Fourier analysis of classical and quantum fields on curved spacetimes previously available mainly for the scalar field on Friedman–Robertson–Walker (FRW) spacetimes are extended to arbitrary vector bundle fields on general spatially homogeneous spacetimes. This is done by developing a rigorous unified framework which incorporates mode decomposition, harmonic analysis and Fourier analysis. The limits of applicability and uniqueness of mode decomposition by separation of the time variable in the field equation are found. It is shown how mode decomposition can be naturally extended to weak solutions of the field equation under some analytical assumptions. It is further shown that these assumptions can always be fulfilled if the vector bundle under consideration is analytic. The propagator of the field equation is explicitly mode decomposed. A short survey on the geometry of the models considered in mathematical cosmology is given and it is concluded that practically all of them can be represented by a semidirect homogeneous vector bundle. Abstract harmonic analytical Fourier transform is introduced in semidirect homogeneous spaces and it is explained how it can be related to the spectral Fourier transform. The general form of invariant bi-distributions on semidirect homogeneous spaces is found in the Fourier space which generalizes earlier results for the homogeneous states of the scalar field on FRW spacetimes.

scholarly journals On Local Fourier Analysis of Multigrid Methods for PDEs with Jumping and Random Coefficients

2019 ◽  
Vol 41 (3) ◽  
pp. A1385-A1413 ◽  
Author(s):  
Prashant Kumar ◽  
Carmen Rodrigo ◽  
Francisco J. Gaspar ◽  
Cornelis W. Oosterlee
Keyword(s):  
Fourier Analysis ◽  
Multigrid Methods ◽  
Random Coefficients ◽  
Local Fourier Analysis
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