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Lp-Asymptotics of Fourier Transform Of Fractal Measures

Electronic Theses of Indian Institute of Science

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Field Value
 
Title Lp-Asymptotics of Fourier Transform Of Fractal Measures
 
Creator Senthil Raani, K S
 
Subject Fractal Measures
Fourier Transform
Hormonic Analysis
Fractal Geometry
Wiener Tauberian type Theorems
Wiener Tauberian Theorem
Mathematics
 
Description One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in Rn. When the support is a smooth enough manifold, an almost complete picture is available. One of the early results in this direction is the following: Let f in Cc∞(dσ), where dσ is the surface measure on the sphere Sn-1 Rn.Then the modulus of the Fourier transform of fdσ is bounded above by (1+|x|)(n-1)/2. Also fdσ in Lp(Rn) for all p > 2n/(n-1) . This result can be extended to compactly supported measure on (n-1)-dimensional manifolds with appropriate assumptions on the curvature. Similar results are known for measures supported in lower dimensional manifolds in Rn under appropriate curvature conditions. However, the picture for fractal measures is far from complete. This thesis is a contribution to the study of asymptotic properties of the Fourier transform of measures supported in sets of fractal dimension 0 < α < n for p ≤ 2n/α.
In 2004, Agranovsky and Narayanan proved that if μ is a measure supported in a
C1-manifold of dimension d < n, then the Fourier transform of μ is not in Lp(Rn) for 1 ≤ p ≤ 2n/d. We prove that the Fourier transform of a measure μ supported in a set E of fractal dimension α does not belong to Lp(Rn) for p≤ 2n/α. As an application we obtain Wiener-Tauberian type theorems on Rn and M(2). We also study Lp-asymptotics of the Fourier transform of fractal measures μ under appropriate conditions and give quantitative versions of the above statement by obtaining lower and upper bounds for the following
limsup L∞ L-k∫|x|≤L|(fdµ)^(x)|pdx
 
Contributor Narayanan, E K
 
Date 2018-06-08T07:38:36Z
2018-06-08T07:38:36Z
2018-06-08
2015
 
Type Thesis
 
Identifier http://etd.iisc.ernet.in/2005/3673
http://etd.iisc.ernet.in/abstracts/4543/G27324-Abs.pdf
 
Language en_US
 
Relation G27324