## Doctoral Theses

### Pompeiu Problem and Analogues of the Weiner-Tauberian Theorem for Certain Homogeneous Spaces.

8-28-1996

8-28-1997

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Abstract (Summary of the Work)

Let G be a connected locally compact unimodular group acting transitively on a locally compact space X. For a function f on X and g â‚¬ G, define of by 9f(x) = fA 9.a), a â‚¬ X. One of the recurring themes in analysis is the question of when a function f in a given function space F(X) will have property that Span{gf : g â‚¬ G} is dense in F(X). If X = R and G = R, the celebrated Wiener-Tauberian theorem answers this question completely for the space L1(R): The span of the translates of fE L1(R) is dense in L1(R) if and only if the Fourier transform f of f is nowhere vanishing on R. If xo E X and Ho = {g â‚¬ G: g.x0 = x0}, then Ho is a closed subgroup of G %3D and the homogeneous space G/Ho can be identified with X under the identification: gHo, ++ g.xo. If, further, Ho is compact and the algebra of compactly supported func- tions on G which are bi-invariant under Ho, is commutative, then the pair (G, Ho) is called a Gelfand pair. In this thesis, we analyse the basic problem stated above for three well known Gelfand pairs: Case (1): Let X = Rn, n ; 2. For G we take the group M(n) of orientation preserving rigid motions of Rn. Any element o of M(n) is given by (T, vo), TE So(n), vo E Rn, its action on Rn being given by o.u = Tv+vo, v E R. Here SO(n), the special orthogonal group, is the collection {A : Aann x nreal matria, AAt = I, det A = 1}. The group law in G is given as follows:(T, v)(S, w) = (TS, Tw + v), (T, v), (S, w) e G. For xo = 0, the origin (0, 0, .., 0) in Rn, the corresponding Ho SO(n). Hence the homogeneous space M(n)/SO(n) can be identified with Rn. It is a well known and easy fact that (M(n), SO(n)) is a Gelfand pair. In Chapter 2, for a function fe L'(X)n IP(X), 1

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