Introduction to hyperfunctions

In summary, the speaker is looking for a tutorial or introduction to hyperfunctions, a topic they recently encountered in their reading. They have found some expensive or out-of-stock textbooks and technical articles, but are still interested in learning more about hyperfunctions.
  • #1
Coelum
97
32
Dear all,
I read about hyperfunctions in Penrose's masterpiece "Road to Reality" and I just met them again (in disguise) in a set of notes on "tricks on computing definite real integrals". Being an engineer with a curiosity for math, I would like to understand a bit more, so I am looking for a tutorial or an introduction to hyperfunctions, possibly with exercises. So far I found only costly or out-of-stock textbooks (like Kaneko's "Introduction to the Theory of Hyperfunctions" - 175$ ! - or Imai's "Applied Hyperfunction Theory" - out of stocks). I also found the original Sato's articles (from Wikipedia), but they are too technical and dry for my background. Any hint?
Thanks,

Francesco
 
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  • #2
Unfortunately, seeing as most of the theory has only been developed in the last 50 years, you can bet that most treatments of the topic will have about that (high) level of technicality. Imai's book would be the least technical and is designed for physicists/engineers, but is quite hard to find. You might just have to find another mathematical curiosity to pursue.
 

1. What are hyperfunctions?

Hyperfunctions are a type of generalized function used in the field of mathematics to model irregular functions that do not satisfy traditional definitions of continuity or differentiability. They were first introduced by Mikio Sato in 1954.

2. How are hyperfunctions different from regular functions?

Regular functions follow the standard definitions of continuity and differentiability, while hyperfunctions are defined as generalized functions that can have singularities and still be considered well-behaved. Hyperfunctions are also defined in terms of their behavior on the boundary of a domain, rather than on the entire domain.

3. What are some applications of hyperfunctions?

Hyperfunctions have been used in various fields of mathematics, including complex analysis, differential equations, and mathematical physics. They have also been applied in signal processing and image reconstruction.

4. How are hyperfunctions represented mathematically?

Hyperfunctions are typically represented using the theory of distributions, which is a generalization of the concept of functions. They can also be represented using Fourier transforms and Laplace transforms.

5. Are there any limitations to using hyperfunctions?

One limitation of using hyperfunctions is that they can be difficult to compute and manipulate, as they are defined in terms of their behavior on the boundary of a domain rather than on the entire domain. Additionally, they may not always have a physical interpretation or meaning in real-world applications.

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