Integral transforms

January 29, 2021 — January 30, 2021

functional analysis
Hilbert space
linear algebra
signal processing
sparser than thou

\[\renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\mmm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\ff}[1]{\mathfrak{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\cc}[1]{\mathcal{#1}}\]

Useful for solving various integral equations and PDEs. Fourier transforms, Laplace transforms, Mellin transforms, Hankel transforms…

Figure 1

Free resource: The transforms and applications handbook / edited by Alexander D. Poularikas. (Poularikas 2000)

1 Fourier transform

See Fourier transforms.

2 Laplace transform

Casual interpretation of the Laplace transform: Generalized Fourier transform. Both more powerful and more difficult, although the nature of the power and the difficulties depends upon where it is to be applied.

\[{\mathcal {L}}\{f\}(s)=\operatorname {E} \!\left[e^{-sX}\right]=\mathbb {E} \!\left[e^{-sX}\right] \]

3 Mellin transform

A transform with “Scale invariance like the Fourier transform has shift invariance”. Sounds fun.

4 Hankel transform

Figure 2

TBD. Covered implicitly in rotational symmetries.

5 References

Adams, and Hedberg. 1999. Function Spaces and Potential Theory.
Bruda. 2022. “Maclaurin Integration: A Weapon Against Infamous Integrals.”
Brychkov, Marichev, and Savischenko. 2019. Handbook of Mellin Tranforms. Advances in Applied Mathematics.
Davies. 2002. Integral Transforms and Their Applications.
Debnath, and Bhatta. 2014. Integral Transforms and Their Applications.
Polyanin, and Manzhirov. 1998. Handbook of Integral Equations.
Poularikas, ed. 2000. The Transforms and Applications Handbook. The Electrical Engineering Handbook Series.
Schiff. 1999. The Laplace Transform: Theory and Applications.
Simon. 2015. Real Analysis. A Comprehensive Course in Analysis 1.0.
Zayed, ed. 2019. Handbook of Function and Generalized Function Transformations.