## Doctoral Theses

### Spectral Analysis and Synthesis for Radial Sections of Homogenous Vector Bundles on Certain Noncompath Riemannian Symmetric Spaces.

2-28-2008

2-28-2009

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Cryptology

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Abstract (Summary of the Work)

We consider two classical theorems of real analysis which deals with translation invariant subspaces of integrable and smooth functions on R respectively. The first one is a theorem of Norbert Wiener [63] which states that if the Fourier transform of a function f âˆˆ L 1 (R) has no real zeros then the finite linear combinations of translations f(x âˆ’ a) of f with complex coefficients form a dense subspace in L 1 (R), equivalently, span{g âˆ— f | g âˆˆ L 1 (R)} is dense in L 1 (R). This theorem is well known as the Wiener-Tauberian Theorem (WTT). The second theorem on spectral analysis on R, due to Laurent Schwartz [56] states that a closed nonzero translation invariant subspace of C âˆž(R) with its usual FrÂ´echet topology contains the map x 7â†’ e iÎ»x for some Î» âˆˆ C. This is equivalent to the statement that if f âˆˆ C âˆž(R) then the closure of the set {W âˆ— f | W âˆˆ C âˆž(R) â€²} in C âˆž(R) contains the map x 7â†’ e iÎ»x for some Î» âˆˆ C, where C âˆž(R) â€² denotes the set of compactly supported distributions on R. We shall call this Schwartzâ€™s theorem. It is well known that the statement above is false for R n if n > 1 (see [31]).We use the terms spectral analysis and spectral synthesis in the sense of Schwartz [56]. We endeavour to study these theorems in the context of homogenous vector bundles on a noncompact rank one Riemannian symmetric space X. We recall that such a space X can be identified with G/K where G is a connected noncompact semisimple Lie group with finite centre having real rank one and K is a maximal compact subgroup of G. This makes X a G-space with canonical G-action. Any function on X can be identified with a right K-invariant function on G and in particular left K-invariant functions on X are K-biinvariant (also called radial) functions on G. In this setup we shall consider the two theorems mentioned above. We shall discuss them one after the other. Wiener-Tauberian Theorem was extended to abelian locally compact groups where the hypothesis is on a Haar integrable function which has nonvanishing Fourier transform on all unitary characters (see [51]). Analogues of this result hold also for many nonabelian Lie groups (see e.g. [27, 43]). On the other hand back in 1955 failure of WTT even for the commutative Banach algebra of integrable radial functions on SL(2, R) was noticed by Ehrenpreis and Mautner in [22]. A simple proof due to M. Duflo of the fact that the WTT based on unitary dual is false for any noncompact semisimple Lie group appears in [43]. This failure can be attributed to the existence of the nonunitary uniformly bounded representations in groups of this class (see [23, 41]).However a modified version of the theorem was established in [22] for radial functions in L 1 (SL(2, R)).

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