E. K. Narayanan

Research Summary:

One of the important problems in Integral geometry is to find out whether certain averages over balls or spheres are sufficient to determine a given continuous function. This can be rephrased as a question of lnjectivity of the associated convolution operator. For example consider the operator $f \rightarrow f* \mu_r,$ where $f$ belongs to some $L^p$ class and $\mu_r$ is the normalised surface measure on the sphere of radius $r > 0$ on $\R^n.$ Few years back, Thangavelu proved that the above operator is injective on $L^p$ as long as $1 \leq p \leq 2n/(n-1).$ Jointly with M. L. Agranovsky we proved a far reaching generalisation of this result, replacing $\mu_r$ with any compactly supported distribution. More precisely if $f * T = 0$ for $f$ in $L^p$ and any compactly supported distribution $T,$ then $f$ vanishes identically as long as $1 \leq p \leq 2n/(n-1).$

Another result related to the spherical means considered above is the so called Support Theorem. Note that if $supp f \subset B(0, R),$ then $f*\mu_r(x) = 0$ if $ r > R + \vert x\vert.$ By a support theorem we mean a converse to this fact under some decay assumptions on $f.$ We completely characterise the spherical harmonic coefficients of the functions with above property and prove a refined version of the well known support theorem of Helgason.


Publications:

  1. The heat kernel and Hardy's theorem on symmetric spaces of noncompact type  Proc. Indian Acad. Sci. 112 (2002), no.2, 321-330 (jointly with S. K. Ray).


Preprints:

  1. $L^p-$ integrability, supports of Fourier transforms and uniqueness for convolution equations (jointly with M. L. Agranovsky).

  2. On support theorems on $ {\Bbb R}^n$.


Conference/Workshops Attended:

Discussion meeting on Harmonic Analysis held at IMSc, Dec 2002.

Visits to other Institutes:

  1. Two short visits to IIT Kanpur in Feb 2003 and April 2003.
  2. One short visit to Stat-Math unit of Indian Statistical Institute, Bangalore, in Dec 2002.

Invited Lectures/Seminars:

  1. Gave a talk at ISI Bangalore in Dec 2002.
  2. Gave a talk at the Discussion meeting in Harmonic Analysis at IMSc in Dec 2002 - Jan 2003.
  3. Gave a talk at IIT Kanpur in Feb 2003.

Other Activities:

    Gave a crash course on Peter-Weyl theorem.


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