Fourier transforms

January 29, 2021 — April 26, 2021

Figure 1

Placeholder.

The greatest of the integral transforms.

1 Incoming

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

Figure 2

2 Useful properties

\(f(x)\,\)
$() = f(x) e^{-i 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?

3 Interpolation

Fourier transforms are useful for cheap interpolations, specifically a magical kind of polynomial basis function.

4 Enveloped Fourier transforms

Figure 3

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 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.

5 References

Dokmanic, and Petrinovic. 2010. Convolution on the \(n\)-Sphere With Application to PDF Modeling.” IEEE Transactions on Signal Processing.
Kausel, and Irfan Baig. 2012. Laplace Transform of Products of Bessel Functions: A Visitation of Earlier Formulas.” Quarterly of Applied Mathematics.
Potts, and Van Buggenhout. 2017. Fourier Extension and Sampling on the Sphere.” In 2017 International Conference on Sampling Theory and Applications (SampTA).
Schaback, and Wu. 1996. Operators on Radial Functions.” Journal of Computational and Applied Mathematics.
Vembu. 1961. Fourier Transformation of the n -Dimensional Radial Delta Function.” The Quarterly Journal of Mathematics.