The Numerical-Valued Fourier Transform in the Two-Sided Operational Calculus

Abstract

The classical and distributional Fourier transform is extended to the ultimate setting in which it can be considered to be numerical-valued. It is extended as a ring isomorphism onto the ring of all , measurable and finite almost everywhere functions under ordinary (pointwise) addition and multiplication of functions. The essential technique needed for this extension is the familiar algebraic procedure (first applied by Mikusiński to the operational calculus in the late 40’s) of imbedding a given ring in a larger ring of fractions, the denominators being nondivisors of zero. Where Mikusiński’s application resulted in a field of one-sided operators, the present application results in a ring of two-sided operators. Beyond this, only classical Fourier analysis is needed, though the extension of the latter to distributions is very useful in identifying many of the operators. A descriptive subtitle for this paper would be Basic definitions and theorems, with applications to be considered later. For reasons of motivation and practical emphasis, the operational calculus is developed in a slightly more restrictive setting where the Fourier transforms are continuous almost everywhere.