*(English)*Zbl 0629.44005

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 ${L}^{1}\left(R\right)\xb7$

An integrable Boehmian F is given by a sequence of quotients $\{{f}_{n}/{\alpha}_{n}\}$ where $\left\{{f}_{n}\right\}$, $n\in N$ is a sequence belonging to ${L}^{1}\left(R\right)$ and $\left\{{\alpha}_{n}\right\}$, $n\in N$ is a delta sequence. Let $F=[{f}_{n}/{\alpha}_{n}]$ be the integrable Boehmian defined by the sequence $\{{f}_{n}/{\alpha}_{n}\}$, $n\in N$, then we denote by ${\widehat{f}}_{n}\left(x\right)$ the Fourier transform of the function ${f}_{n}\left(t\right)$. The Fourier transform of the integrable Boehmian $F=[{f}_{n}/{\alpha}_{n}]$ is the limit function of the sequence ${\widehat{f}}_{n}\left(x\right)$ 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($\lambda )$ defined on the interval [0,1], with values in the space of integrable Boehmians, is such that the derivative F’($\lambda )$ 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.

##### MSC:

44A40 | Calculus of Mikusiński and other operational calculi |

42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |

46F12 | Integral transforms in distribution spaces |