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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 /α n } where {f n }, nN is a sequence belonging to L 1 (R) and {α n }, nN is a delta sequence. Let F=[f n /α n ] be the integrable Boehmian defined by the sequence {f n /α n }, nN, then we denote by f ^ n (x) the Fourier transform of the function f n (t). The Fourier transform of the integrable Boehmian F=[f n /α n ] is the limit function of the sequence 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(λ) defined on the interval [0,1], with values in the space of integrable Boehmians, is such that the derivative F’(λ) 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:
44A40Calculus of Mikusiński and other operational calculi
42A38Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
46F12Integral transforms in distribution spaces