## 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 $$L^ 1(R).$$
An integrable Boehmian F is given by a sequence of quotients $$\{f_ n/\alpha_ n\}$$ where $$\{f_ n\}$$, $$n\in N$$ is a sequence belonging to $$L^ 1(R)$$ and $$\{\alpha_ n\}$$, $$n\in N$$ 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 $$\hat f_ n(x)$$ the Fourier transform of the function $$f_ n(t)$$. The Fourier transform of the integrable Boehmian $$F=[f_ n/\alpha_ n]$$ is the limit function of the sequence $$\hat f_ n(x)$$ 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.
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

### Citations:

Zbl 0495.44006; Zbl 0226.46001
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