Generalized n-dimensional hilbert transform and applications

Abstract

Let be the Schwartz test function space, consisting of infinitely differentiable functions φ defined over Rⁿ such that φ^k(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):

• 46F12
• 44A15