Seminar: Range of Certain Convolution Operators and Reconstruction from Local Averages

Abstract

Let μ be a compactly supported regular Borel measure on a locally compact abelian group G and let C(G) denote the set of all complex valued continuous functions on G. We analyse the convolution operator C μ : C(G) −→ C(G) defined by C μ (f )(x) = (f μ)(x) := f (x − y)dμ(y). For a given function g ∈ C(G), we are interested in knowing whether there exists f such that C μ (f ) = g. This is equivalent to characterizing the range of the operator C μ . We are also interested in finding a solution f when g is in the range of C μ . We show that when μ is a compactly supported discrete Borel measure, the range of the operator C μ is C(R n ) for n = 1, 2. Further, we show that when μ is absolutely continuous with respect to the Lebesgue measure and has a density function which is a finite linear combination of indicator functions of intervals, then the range of C μ is C 1 (R). In this case, for g ∈ C 1 (R), we explicitly construct a solution f in C(R) such that C μ (f ) = g. We observe that even if g ∈ L 1 (R) ∩ C 1 (R), μ(λ) = 0 and g(λ) = 0, there is a function f ∈ C(R) such that C μ (f ) = g. We also consider certain aspects of local average sampling and reconstruction over spline spaces of polynomial growth.

Date and Time : 23-01-2017 10:00

Venue : Seminar Hall, 219 @ CET

Speaker : P. Devraj, Anna University

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