Fourier transforms



Placeholder.

The greatest of the integral transforms.

Misc

I especially need to learn about Fourier transforms of radial functions.

Useful properties

\(f(x)\,\)
\(\hat{f}(\nu) = \int f(x) e^{-i \nu x}\, dx\)
Linearity: \(a\cdot f(x) + b\cdot g(x)\,\)
\(a\cdot \hat{f}(\nu) + b\cdot \hat{g}(\nu)\,\)
Time shift \(f(x - a)\,\)
\(e^{- i a \nu} \hat{f}(\nu)\,\)
Frequency shift \(f(x)e^{iax}\,\)
\(\hat{f}(\nu - a)\,\)
Dilation \(f(a x)\,\)
\(\frac{1}{|a|} \hat{f}\left( \frac{\nu}{a} \right)\,\)
Duality \(\hat{f}(x)\,\)
\(2\pi f(-\nu)\,\)
\(\frac{d^n f(x)}{dx^n}\,\)
\((i\nu)^n \hat{f}(\nu)\,\)
\(x^n f(x)\,\)
\(i^n \frac{d^n \hat{f}(\nu)}{d\nu^n}\)
Convolution \((f * g)(x)\,\)
\(\hat{f}(\nu) \hat{g}(\nu)\,\)
\(f(x) g(x)\,\)
\(\frac{1}{2\pi}\left(\hat{f} * \hat{g}\right)(\nu)\,\)
Hermitian symmetry. For\(f(x)\) purely real
\(\hat{f}(-\nu) = \overline{\hat{f}(\nu)}\,\)
For \(f(x)\) purely real and even
\(f(\nu)\) is purely real even functions.
For \(f(x)\) purely real and odd
\(f(\nu)\) are purely imaginary and odd.
For\(f(x)\) purely imaginary
\(\hat{f}(-\nu) = -\overline{\hat{f}(\nu)}\,\)
Complex conjugation \(\overline{f(x)}\)
\(\overline{\hat{f}(-\nu)}\)
\(f(x) \cos (a x)\)
\(\frac{\hat{f}(\nu-a)+\hat{f}(\nu+a)}{2}\)
\(f(x)\sin( ax)\)
\(\frac{\hat{f}(\nu-a)-\hat{f}(\nu+a)}{2i}\)
\(e^{-\alpha x^{2}}\)
\({\sqrt{\frac {\pi }{\alpha }}}\cdot e^{-{\frac {\nu ^{2}}{4\alpha}}}\)
\(e^{-i\alpha x^{2}}\)
\({\sqrt{\frac {\pi }{\alpha }}} \cdot e^{i(\frac {\nu ^{2}}{4\alpha}-\frac {\pi }{4})}\)
\(f(x)e^{-\alpha x^{2}}\)
…anything useful?

Enveloped Fourier transforms

If you multiply your function by an envelope or tapering function or window or whatever before taking a Fourier transform, then this is no longer strictly an FT; but I wonder which features we can relate this back to FTs?

This is heavily studied in the context of time frequency analyses, but I have not found a reference for all the results I need, so I will derive a couple of my own here.

References

Dokmanic, I., and D. Petrinovic. 2010. “Convolution on the $n$-Sphere With Application to PDF Modeling.” IEEE Transactions on Signal Processing 58 (3): 1157–70. https://doi.org/10.1109/TSP.2009.2033329.
Kausel, Eduardo, and Mirza M. Irfan Baig. 2012. “Laplace Transform of Products of Bessel Functions: A Visitation of Earlier Formulas.” Quarterly of Applied Mathematics 70 (1): 77–97. https://doi.org/10.1090/S0033-569X-2011-01239-2.
Potts, Daniel, and Niel Van Buggenhout. 2017. “Fourier Extension and Sampling on the Sphere.” In 2017 International Conference on Sampling Theory and Applications (SampTA), 82–86. Tallin, Estonia: IEEE. https://doi.org/10.1109/SAMPTA.2017.8024365.
Schaback, Robert, and Z. Wu. 1996. “Operators on Radial Functions.” Journal of Computational and Applied Mathematics 73 (1): 257–70. https://doi.org/10.1016/0377-0427(96)00047-7.
Vembu, S. 1961. “Fourier Transformation of the n -Dimensional Radial Delta Function.” The Quarterly Journal of Mathematics 12 (1): 165–68. https://doi.org/10.1093/qmath/12.1.165.

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