Introduction to Hyperfunctions and Their Integral Transforms

This textbook presents an elementary introduction to generalized functions by using Sato's approach of hyperfunctions which is based on complex function theory. This very intuitive and appealing approach has particularly great computational power.

The concept of hyperfunctions and their analytic properties is introduced and discussed in detail in the first two chapters of the book. Thereafter the focus lies on generalizing the (classical) Laplace, Fourier, Hilbert, Mellin, and Hankel transformations to hyperfunctions. Applications to integral and differential equations and a rich variety of concrete examples accompany the text throughout the book.

Requiring only standard knowledge of the theory of complex variables, the material is easily accessible for advanced undergraduate or graduate students. It serves as well as a reference for researchers in pure and applied mathematics, engineering and physics.

• DEX
• Fourier transform
• Hyperfunction
• Integral equation
• computation
• distribution
• equation
• form
• function
• functions
• generalized function
• integral
• integral transform
• variable

From the reviews:

“The intended audience of the book is advanced undergraduate students, graduate students, and research professionals in mathematical sciences. The book can serve as a research reference or a supplement to courses or seminars. … this book provides an excellent presentation of integral transforms of hyperfunctions with examples of applications to integral equations and ordinary and partial differential equations. … would be successful as a senior undergraduate- or graduate-level text in mathematics, physics, and engineering, as well as a research reference for professionals in those fields.” (Lokenath Debnath, SIAM Review, Vol. 53 (1), 2011)

“This is an introductory and elementary textbook about Sato’s hyperfunctions in one variable and some of their integral transforms. … mainly addressed to applied mathematicians, physicists and engineers, it may also be useful for pure mathematicians seeking a first encounter with Sato’s beautiful conception of generalized functions. … The only prerequisites for the reader are elementary notions from complex function theory of one variable and some familiarity with the classical Laplace transform. This makes the material of the book accessible even to undergraduate students.” (Jasson Vindas, Mathematical Reviews, Issue 2012 a)

“This monograph aims at introducing the theory of hyperfunctions and some of their integral transforms to a wide range of readers, since the author recognized that the noble idea of hyperfunction was known to only few mathematicians.” (Dohan Kim, Zentralblatt MATH, Vol. 1201, 2011)

• La Neuveville, Switzerland

  Urs Graf

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