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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 \(L^ 1(R).\)

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.

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 |