## Date of Submission

7-28-2008

## Date of Award

7-28-2009

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Doctoral Thesis

## Degree Name

Doctor of Philosophy

## Subject Name

Mathematics

## Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

## Supervisor

Bagchi, Somesh Chandra (TSMU-Kolkata; ISI)

## Abstract (Summary of the Work)

Classical Fourier analysis derives much of its power from the fact that there are three function spaces whose images under the Fourier transform can be exactly determined. They are the Schwartz space, the L2 space and the space of all C âˆž functions of compact support. The determination of the image is obtained from the definition in the case of Schwartz space, through the Plancherel theorem for the L 2space and through the Paley-Wiener theorem for the other space.In harmonic analysis of semisimple Lie groups, function spaces on various restricted set-ups are of interest. Among the multitude of these spaces it is again the spaces analogous to the three spaces above for which characterization of images under Fourier transform has been possible. Having neither the advantage of the duality nor the well behaved characters as the Euclidean set-up, the determination of images has been hard work in all the situations here- leading to the Schwartz space isomorphism theorems, the Paley-Wiener theorems and the Plancherel theorems. Some of these results have been reworked in recent years resulting in simpler approaches and redefining the interrelationships of these results. This the context of the present thesis.Our set-up is a connected, non-compact, semisimple real Lie group G having finite center and K a maximal compact subgroup of G. A main inspiration for our work is J-P. Ankerâ€™s [2] proof of Schwartz space isomorphism under the Fourier transform for bi-K-invariant functions on G. Unlike the earlier proofs of this result, this beautiful proof relies on the Paley-Wiener theorem and takes no recourse to the asymptotics of elementary spherical functions due to Harish-Chandra except, indirectly, for what is involved in the Paley-Wiener theorem. Since a proof of the Paley-Wiener theorem had 1 already been found that did not use the Schwartz space isomorphism theorem as well, Ankerâ€™s proof thus scripted an â€˜elementaryâ€™ development of Harmonic Analysis of bi-K-invariant functions.It is in the above spirit that we take up our first function space, the L p -Schwartz space S p Î´ (X) (0 < p â‰¤ 2) of a given (left) K-type Î´ on the symmetric space X = G/K under the assumption that G/K is of real rank-1. The relevant Fourier transform here is the Î´-spherical transform. In characterizing the image of the Î´-spherical transform, we do not attempt to adopt the arguments of Anker as suggested in [2]. Instead we exploit the Kostant polynomials to reduce the problem to the bi-K-invariant case and use Ankerâ€™s result thereafter. Again this provides arguments relying on the Paley-Wiener theorem to prove our result which is a part of the Eguchi-Kowata theorem [9] (where they established the isomorphism for S p (X) without the restriction of the left type).

## Control Number

ISILib-TH313

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

## DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

## Recommended Citation

Jana, Joydip Dr., "Isomorphism of Schwartz Spaces Under Fourier Transform." (2009). *Doctoral Theses*. 42.

https://digitalcommons.isical.ac.in/doctoral-theses/42

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842818