Fourier Analysis on GL(n,R).

Area of Science:

• Mathematics
• Harmonic Analysis
• Group Theory

Background:

• Fourier analysis on matrix groups is a complex field.
• Understanding group representations and analytic continuation of zeta-functions are key challenges.

Purpose of the Study:

• To decompose the additive Fourier operator using group representation theory.
• To analytically continue zeta-functions defined on the general linear group GL(n,R).

Main Methods:

• Utilizing group representation theory for operator decomposition.
• Applying techniques for the analytic continuation of zeta-functions.

Main Results:

• The additive Fourier operator decomposition was achieved.
• Analytic continuation of specific zeta-functions was performed.
• Generalized Gamma functions emerged as a unifying element.

Conclusions:

• Generalized Gamma functions provide a natural framework for solving problems in Fourier analysis on GL(n,R).
• This work connects group representation theory and analytic number theory through these functions.