Solution to Some Singular Integro-Differential Equations in the Space of Semi-Almost Periodic Distributions
by
J. N. Pandey
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada
Coauthors: Fayez Seiffedine

Let a(t) be an infinitely differentiable almost periodic function in the sense of Bohr and f be a semi-almost periodic distribution defined on the real line and let H be the operator of Hilbert transform defined on these spaces. The space of semi-almost periodic distribution that we choose contains the Schwartz space of almost periodic distributions as well as regular distributions generated by the space of simple functions defined on the real line.

Our objective is to solve the following singular integro-differential equations in the space of semi-almost periodic distributions defined on R.

y'+ a(t) Hy = f (1) y''+ a(t) Hy = f (2)

We have solved these problems when a(t) is a constant. The case when a(t) not a constant is an open problem.

We have also considered the generalisations of these problems when a(t) is an n × n matrix whose elements are infinitely differentiable almost periodic functions on R and f is an n-column vector with almost periodic or semi-almost periodic elements. The symbol y will have similar meaning. The case when a(t) is a constant n × n diagonalizable matrix has been solved by us but the general case when a(t) is not a constant matrix is still an open problem.

We will also discuss the wavelet transform of a. p. distributions or semi-almost periodic distributions.

Date received: February 19, 2003