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Henstock-Kurzweil Fourier transforms. (English) Zbl 1037.42007
For functions \(f:\mathbb R \to \mathbb R\) the Fourier transform given by \(\widehat f(s)=\int_{-\infty}^{\infty} e^{-isx}f(x)dx\) is considered where the integral is the non-absolutely convergent Henstock-Kurzweil integral.
The author follows the main properties of the Fourier transform known for the case of Lebesgue integrable functions \(f\) for the case of functions which are Henstock-Kurzweil integrable (Riemann-Lebesgue lemma, differentiation of the Fourier transform, transform of derivatives, convolutions, properties of \(\widehat f\), etc.).
In any respect the Henstock-Kurzweil integration changes the situation dramatically, e.g., in the case of the Riemann-Lebesgue lemma.
The paper seems to be a first systematic study of the Fourier transform based on the concept of the Henstock-Kurzweil integral and starts the possible research in this field for delicate problems of the Fourier transform of functions integrable in this sense.

MSC:
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
26A39 Denjoy and Perron integrals, other special integrals
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