## Date of Submission

2-28-1999

## Date of Award

2-28-2000

## 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)

The uncertainty principles of Harmonic Analysis say that: a nonzero func- tion and its Fourier transform cannot both be sharply concen- trated. After the initial work on this phenomenon in 1920s, the last two decades witnessed a spurt of activity in this direction (we refer the reader to a very readable survey [FS]). One may notice two broad phases in this activity, the first concentrating on R where the notion of concentration is given different formulations to see whether the phenomenon still holds. In the later phase R is replaced by other commutative or noncommutative groups, or more generally by homogeneous spaces to see which uncertainty principles remain valid.In this thesis our main objective is to get analogues of the following theorem due to Cowling and Price on some classes of Lie groups.Theorem 1.0.1 Suppose f : R -C be a measurable function and sat- isfiesIS(x)Pdr < o0,< (ii) fa egbry|f(y)|;dy < 0o,where min(p, q) < 00, a, b > 0 and f(y) = SR f(x)e-2rizy dr is the Fourier transform of f. If ab 2 1 then f = 0 almost everywhere and if ab < 1 then there ezist nonzero functions satisfying the above conditions. For the proof of the theorem see CP). Their motivation for the result is a classical result due to Hardy which usen L" norm instead of P and L norms; namely, if f :R - C is a measurable function such that IS(x)| s Ceer, fulis Ce t with a, b,C & gt; 0, then = 0 almost everywhere if ab > 1. f(z) = Ceir ab =1 and the case ab (see (HJ).Whereas the res ult of Hardytates that f and f cannot be very rapidly decreasing pointwise, that of Cowling and Prices aerta mere; it says that both f and f cannot decay very rapidly on an average. Barrizng the case ab - I (erns to be something special to Euclidean spaces), we see that Hardy theorem follows from that of Cowling and Price, as experted. However one of our results (Theorem 1.1.4) shows that for ;, the case ab & gt; 1 of the Cowling rice theorem can be obtained from Hardy thearem in an elementary way, without the sulete modification of the Phragmen-LÃ¡ndekiff theorem originally employed. We should mention that because of the cae ab = 1, these two theorems still stand as independent theorems at least on Rn. Though these results on Ruses complex variable methods (mainly Phragnen-Lindelidr theorem) via entire extension of the Fourier transform, their extenskons to; need only analysis of ve complex variable.The complex analytic techniques used in Cowling-Price theorem motivates us to look at nemisimple Lie groups. The decay of the matrix coefficients of the Principal series representations prepare the ground for an approach similar to what in done on the Euclidean spaces. There are, how- ever, two serious oletaclon in the way, The first one ls the existence of the discrete series and the second one is caused by the zeros of the real analytic function appearing in the Plancherel theorem. While we g the first obstacle, using the main kiden of CSS), the second obliges us to restrict ourselves to the rank one case. The method of analytic continuation is also found to work for some group outside of the semisimple class.

## Control Number

ISILib-TH177

## 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

Ray, Swagato Kumar Dr., "Uncertainity Principles on Some Lie Groups." (2000). *Doctoral Theses*. 420.

https://digitalcommons.isical.ac.in/doctoral-theses/420

## 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:28843819