## Doctoral Theses

### Fourier Transforms of Very Rapidly Decreasing Functions on Certain Lie Groups.

2-28-1996

2-28-1997

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)

#### Abstract (Summary of the Work)

Recall that for a function f Ïµ L1(Rn ), its Fourier transform fÌ‚ is definedby: fÌ‚ (Æ¹) = Êƒ Rnf(x)ei(Æ¹,x)dx ( 0.1.1)where (.,.) denotes the standard inner product on Rn and dr the Lebesgue measure on Rn. A celebrated theorem of L. Schwartz asserts that a function f on Rn is rapidly decreasing (or in the Schwartz class ) if and only if its Fourier transform is rapidly decreasing . In sharp contrast to Schwartz s theorem, is a result due to Hardy ([18) which says that Êƒ and fÌ‚ cannot both be very rapidly decreasing . More precisely, if | f (x) â‰¤ Ae-ox2 and | fÌ‚ (Æ¹) â‰¤ Be-Î²Æ¹2 for some positive constants Î± Î² and aÎ² >Â¼,then f = 0. Hardy s theorem can also be viewed as a sort of uncertainty principle. Roughly speaking, various uncertainty principles, including the celebrated Heisenberg uncertainty principle, say that a non-trivial function and its Fourier transform cannot be simultaneously concentrated . Depending on the definition of concentration , we get a host of uncertainty principles (-see [2), (3), (4), (7), (9), [24), [25), [28], [29), [30), [33), [38), (40] etc ). Clearly, Hardy s theorem belongs to this class of results where concentration is measured in terms of rate of decay of f and fÌ‚ at infinity. Some of the uncertainty principles seem to be valid even in very abstract situations. For instance, in [5), M. F. E. De Jeu has shown that the uncertainty principle due to Donoho and Stark ((7) is valid whenever one has an integral operator for which a Plancherel theorem holds. For an account of uncertainty principles and their connections with physics etc see (8) or [34].Since the theorem of Schwartz is of fundamental importance in harmonic anal- ysis, there is a whole body of literature (-see for instance [35], p.151 and (43)-) devoted to generalizing this result to other Lie groups. However, as far as we are a ware, until very recently no systematic attempt was made to generalize Hardy s theorem in the context of harmonic analysis on Lie groups. In this thesis, we shall give generalizations of Hardy s theorem to the Heisenberg group, the n-dimensional Euclidean motion group and a sub class of noncompact semi-simple Lie groups.Let G be a locally compact, unimodular group satisfying the second axiom of countability. Moreover, assume that Äœ, is postiliminaire. (For the precise definitions, the reader may refer to (6), pp.303 and (23), pp.196.) Let dmg denote the Haar measure on Äœ,. Let Äœ, be its unitary dual, i.e. the set of equivalence classes of continuous, irreducible, unitary representations of Äœ,. Given fe L (G), we define the group Fourier transform f of f by:fÌ‚(Ï€) = Ï€(f) = Êƒ Gf(x)Ï€(x)dmG(x), Ï€â‚¬ Äœ. ( 0.1.2) %3D(For Ï€ Îµ Äœ, let H, be the underlying Hilbert space on which Äœ, acts. The above integral is to be interpreted suitably as an element of B(HÏ€), the collection of bounded linear operators on HÏ€.) Then, by the abstract Plancherel theorem, there exists a measure structure and a unique positive measure u on Äœ, such that for f Îµ L1(G)L (G),Êƒ Gâ”‚f(x)â”‚2dmG(x) = Êƒ Äœ,tr (Ï€(f)'Ï€(f)dÂµ(Ï€).

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