K. Parthasarathy

On weak spectral synthesis

Weak spectral synthesis fails in the group algebra and the generalised group algebra of any non compact locally compact abelian group and also in the Fourier algebra of any infinite compact Lie group.

Feichtinger's Segal algebra on homogeneous spaces

Let K be a compact subgroup of a locally compact group G. We extend to the context of homogeneous spaces, G/K, the definition of Feichtingers Segal algebra. The functorial properties of the Segal algebra are proved.

Wiener Tauberian theorems for vector-valued functions

Different versions of Wieners Tauberian theorem are discussed for the generalized group algebra L 1 ( G , A ) (of integrable functions on a locally compact abelian group G taking values in a commutative semisimple regular Banach algebra A ) using A -valued Fourier transforms. A weak form of Wieners Tauberian property is introduced and it is proved that L 1 ( G , A ) is weakly Tauberian if and only if A is. The vector analogue of Wieners L 2 -span of translates theorem is examined.

Phase space frames and frame operators

An explicit, simple characterization of those operators that arise as Weyl–Heisenberg frame operators on L2(ℝn) is obtained in terms of the concept of tile vertices of an operator. Generalizations of these frames to abstract Hilbert spaces are introduced and these and their frame operators are characterized in terms of Weyl–Heisenberg frames and frame operators in Euclidean spaces. Appropriate window versions of these concepts and results are also presented.