Jan Mikusinski developed an operational calculus which is relevant for solving differential equations. His operational calculus is based upon an algebra of the convolution of functions with respect to the Fourier transform. From the convolution product he goes on to define what in other contexts is called the field of fractions or a quotient field. These ordered pairs of functions Mikusinski calls operators.

Algebraic Definition of Operators

The set of functions and the operation of convolution define a commutative ring. Any ring without divisors of zero can be extended to a quotient field or field of fraction such that

• b/a = d/c if and only if b*c = a*d.
• a = (a*k)/k for any k not equal to the zero of the ring.

The convolution quotient of a function with itself; i.e., f/f; corresponds to the Dirac delta function, δ(t), the unit element of the set of generalized functions.

(To be continued.)