Date of Submission

2-28-1996

Date of Award

2-28-1997

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-Bangalore)

Supervisor

Sitaram, Alladi (TSMU-Bangalore; ISI)

Abstract (Summary of the Work)

Recall that for a function f ϵ L1(Rn ), its Fourier transform f̂ is definedby: f̂ (ƹ) = ʃ Rnf(x)ei(ƹ,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 ʃ and f̂ cannot both be very rapidly decreasing . More precisely, if | f (x) ≤ Ae-ox2 and | f̂ (ƹ) ≤ Be-βƹ2 for some positive constants α β and aβ >¼,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 f̂ 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 Ĝ, 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 Ĝ,. Let Ĝ, be its unitary dual, i.e. the set of equivalence classes of continuous, irreducible, unitary representations of Ĝ,. Given fe L (G), we define the group Fourier transform f of f by:f̂(π) = π(f) = ʃ Gf(x)π(x)dmG(x), π€ Ĝ. ( 0.1.2) %3D(For π ε Ĝ, let H, be the underlying Hilbert space on which Ĝ, acts. The above integral is to be interpreted suitably as an element of B(Hπ), the collection of bounded linear operators on Hπ.) Then, by the abstract Plancherel theorem, there exists a measure structure and a unique positive measure u on Ĝ, such that for f ε L1(G)L (G),ʃ G│f(x)│2dmG(x) = ʃ Ĝ,tr (π(f)'π(f)dµ(π).

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:28842928

Control Number

ISILib-TH224

Creative Commons License

Creative Commons Attribution 4.0 International 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

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