The Boehmians have been introduced by J. Mikusiński and the author [C. R. Acad. Sci., Paris, Sér. I, 293, 463-464 (1981; Zbl 0495.44006)]. In this paper the author, using Burzyk’s method, defines the Fourier transform for integrable Boehmians. The Fourier transform of an integrable Boehmian is a continuous function and has all basic properties of the Fourier transform in
An integrable Boehmian F is given by a sequence of quotients where , is a sequence belonging to and , is a delta sequence. Let be the integrable Boehmian defined by the sequence , , then we denote by the Fourier transform of the function . The Fourier transform of the integrable Boehmian is the limit function of the sequence in C(R). The author proves that the Fourier transform of every integrable Boehmian exists and is a continuous function on R. He proves basic properties and an inversion formula for this Fourier transform.
A consequence of the author’s results is the important theorem proved by S. Rolewicz [Metric linear spaces (1972; Zbl 0226.46001)]: If a function F( defined on the interval [0,1], with values in the space of integrable Boehmians, is such that the derivative F’( exists and is equal to 0 at each point, then F is a constant function. It is interesting that many mathematicians tried to prove a similar theorem for the Mikusiński operators but without success.