Publication Cover
Applicable Analysis
An International Journal
Volume 20, 1985 - Issue 3-4
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Original Articles

Generalized n-dimensional hilbert transform and applications

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Pages 221-235 | Received 10 Feb 1985, Published online: 20 Jan 2011
 

Abstract

Let be the Schwartz test function space, consisting of infinitely differentiable functions φ defined over Rn such that φk(x) belongs to for each k=0, 1, 2, 3,…. We shall introduce the space of all test functions of the form.

where each and k is finite, and the function
We shall prove that with the sub- space topology induced on it by the topology of is dense in The Hilbert transform of test functions in is defined by:
It is seen that the operator is a linear embedding and satisfies
The Hilbert transforn of test functions is then defined by
where is a sequence in converging to φ in the sense of the convergence in . It is seen that is a homeomorphism with its inverse given by
The Hilbert transforn Hf of is then defined by
It is seen that the generalized Hilbert transform defined by the above relation is a linear isomorphism from onto itself and satisfies Applications of our result to solve some singular integral equations are also shown.

AMS(OMS):

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