## Fourier transform on integrable Boehmians.(English)Zbl 1191.42003

The authors first establish sufficient conditions for a function to be in $$(L^1(\mathbb{R}))^{\wedge}$$, the space of the Fourier transforms of integrable functions on $$\mathbb{R}$$. Moreover, two characterizations for continuous functions vanishing at $$\infty$$ and which belong to $$(L^1(\mathbb{R}))^{\wedge}$$ are proved and used to give some examples. Finally, they characterize the continuous functions which are Fourier transforms of functions in the Boehmian space introduced by D. Nemzer [Appl. Anal. Discrete Math. 1, 172–183 (2007; Zbl 1199.46096)]. This allows them to answer a question raised by Nemzer in that paper.

### MSC:

 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 44A40 Calculus of Mikusiński and other operational calculi

Zbl 1199.46096
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### References:

 [1] DOI: 10.4064/cm108-2-8 · Zbl 1274.42018 [2] Atanasiu D., Int. J. Math. Math. Sci. 16 pp 2579– (2005) · Zbl 1099.46028 [3] Karunakaran V., Fract. Calc. Appl. Anal. 5 pp 181– (2002) [4] DOI: 10.4064/cm102-1-3 · Zbl 1079.46029 [5] Katznelson Y., An introduction to Harmonic Analysis (1976) · Zbl 0352.43001 [6] Mikusinski P., Japan. J. Math. pp 159– [7] DOI: 10.1216/RMJ-1987-17-3-577 · Zbl 0629.44005 [8] DOI: 10.1007/BF01903334 · Zbl 0652.44005 [9] Nemzer D., Appl. Anal. Discrete Math. 1 pp 172– (2007) · Zbl 1199.46096 [10] Roopkumar R., Int. J. Math. Game theory Algebra. 13 pp 465– (2003) [11] Rudin W., Functional Analysis, 2. ed. (1991) · Zbl 0867.46001
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