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Fourier transform for integrable Boehmians. (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 ${L}^{1}\left(R\right)·$

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

Reviewer: B.Stanković

##### 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