| Summary: | We generalize the classic Fourier transform operator <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">F</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula> by using the Henstock–Kurzweil integral theory. It is shown that the operator equals the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>H</mi> <mi>K</mi> </mrow> </semantics> </math> </inline-formula>-Fourier transform on a dense subspace of <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="script">L</mi> <mi>p</mi> </msup> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>. In particular, a theoretical scope of this representation is raised to approximate the Fourier transform of functions on the mentioned subspace numerically. Besides, we show the differentiability of the Fourier transform function <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="script">F</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> under more general conditions than in Lebesgue’s theory. Additionally, continuity of the Fourier Sine transform operator into the space of Henstock-Kurzweil integrable functions is proved, which is similar in spirit to the already known result for the Fourier Cosine transform operator. Because our results establish a representation of the Fourier transform with more properties than in Lebesgue’s theory, these results might contribute to development of better algorithms of numerical integration, which are very important in applications.
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