## Date of Submission

2-28-1997

## Date of Award

2-28-1998

## 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)

In their celebrated study of Harmonic analysis on semi-simple Lie groups Ehrenpreis and Mautner [E-M] noticed that the analogue of the claasical Wiener Tauberian theorem resting on the unitary dual does not hold for semisimple Lle groups. A simple proof of this fact due to M. Duflo appears in (H). Ehrenpreis and Mautner went on in (E-M] to formulate the problem on the commutative Banach algebra of the SO2(R)-biinvariant functions in L1(SL2(R))1, and obtained two different versions of the theorem involving, this time, the dual of the Banach algebra which includes, beside the unitary dual of G, a part of the non-unitary dual as well. They considered the problem of a single function f generating the Banach algebra of bi-invariant functions as a closod ideal. Among the many articles inspired by [E-M] in the interven- ing years (S1], [S2]; and [B-W] are of particular importance in our context. In (S1), Sitaram proved one of the theorems of Ehrenpreis-Mautner, in the setting of a connected semisimple Lie groups G with a finite centre in place of SL (R). In [S2] he considers a function s in L1(SL2(R)/SO2(R)) which is of finite type under left SO;(R) action to get a sufficient, conditions for f to generate a dense aubspace of L1(SL2(R)/SO:(R)) under left convolution by L1(SL2(R)) functions. In a recent paper of Benyamini and Weit [B-W), auch sufficient conditions are obtained on a family of bi-invariant functions instead of on a single function, so that the closed ideal generatod by them la the entire algebra of bi-invariant functions in L1(SL2(R)).In this thesis we obtain Wiener Tauberian (W-T) theorens for the whole space L1(SL2(R)) as well as for LP(SL(R)) for 1 SpS 2 and a W-T theorem for IP(G/K) when G ia a connected semisimple Lie group of real rank one with finite centre. Along the way, we examine some related questions as also some of the earlier proofs of the available W-T theorems. Our treatment relies heavily on the results of (B-W] and on the characterization of Fourier transforms of the Schwartz spaces CP(G) obtained by Trombi (T) and Barker [Bal.Throughout this thesis G will denote a semisimple Lie group and K will be a maximal compact subgroup of G. We begin with G = SL2(R) and K = SO2(R). We denote the characters of K by Xn, n â‚¬ Z. A complex valued function f on G is said to be of left (resp. right) K-type n if f(ka) = Xn(k)S(x) %3D (resp. f(zk) = Xn(k)S(x)), for all k e K and a â‚¬ G. For a class of functions F on G (e.g. LP(G)), Fn will denote the corresponding subclass of functions of right type n while Fm,n will comprise funtions in Fn which are also of left type m. The principal series representations of SL2(R) are parametrized by (o, A), where o e M and A e C, and the discrete series ropresontations are parametrized by the integers. Principal and discrete parts of the Fourier transform of a function f will be denoted by fH and fB respectively. Unless mentioned otherwise p will lie in (1, 2).

## Control Number

ISILib-TH283

## 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

Sarkar, Rudra Pada Dr., "Wiener Tauberian Theorems on Semisimple Lie Group." (1998). *Doctoral Theses*. 344.

https://digitalcommons.isical.ac.in/doctoral-theses/344

## 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:28843447