MMfE - Examples of Fourier Transform

All Exercises (with Solutions) containing the “Fourier_transform”, “is_Fourier_transformable”, “inverse_Fourier_transform” or “properties_of_Fourier_transform”, tag:

• MMfE ~ exam_2020_06_10 - with solutions
• MMfE ~ exam_2020_06_24 - with solutions
• MMfE ~ exam_2020_07_15 - with solutions
• MMfE ~ exam_2021_02_17 - with solutions

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General Solution

• If a function belongs to at least the first Lobsgue space, so $f(x)\in L^1$ (or $L^2$, $L^3$, $\dots$), then we can say that it’s Fourier Transform exist, so we should assert:

$${\int }_{-\infty }^{+\infty }\left|f(t)\right|dt<\infty$$

We say that the function $f(t)$ is Lebesgue-measurable (Wikipedia)

• Then we can proceed with the actual transformation, usually done with the definition of Fourier Transformation:

$$F(w)={\int }_{-\infty }^{+\infty }f(t)\cdot e^{-iwt}dt$$

• The inverse Fourier Transform is given by:

$$f(t)=\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }F(w)\cdot e^{iwt}dw$$

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~Ex.: 2020_06_10 Point 2.

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~Ex.: 2020_06_10 Point 3.

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~Ex.: 2020_06_24 Point 3.

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~Ex. 2020_07_15 Point 3.

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Transclude of MMfE-~-exam_2021_02_17---with-solutions#ex-2021_02_17-point-4