Invariant mean value property and harmonic functions. (English) Zbl 1097.46024

Kim, Jinman; Wong, M. W.

Complex Variables, Theory Appl. 50, No. 14, 1049-1059 (2005).

The authors attempt to obtain conditions on functions \(\sigma\) and \(u\) on \(\mathbb{R}^n\) such that, if \(u\) is given by the convolution of \(\sigma\) and \(u\), then \(u\) is harmonic on \(\mathbb{R}^n\). Fourier hyperfunctions are defined in the sequel.
For a generalization of the Fourier transform and some other generalizations of well-known integral transforms, the reviewer refers to his publications [J. M. C.Joshi, Integral Transforms Spec.Funct.15, No. 2, 117–127 (2004; Zbl 1063.44004); J.Nat.Phys.Sci.11, 65–78 (1997; Zbl 0976.44001)]. These generalizations have been named Dr.S.M.Joshi integral transform after the reviewer’s respected father.

Reviewer: J. M. C. Joshi (Uttaranchal)

MSC:

46F12	Integral transforms in distribution spaces
46F15	Hyperfunctions, analytic functionals
30D15	Special classes of entire functions of one complex variable and growth estimates

Keywords:

Fourier transform; heat kernel; Fourier hyperfunctions; convolution transform

Citations:

Zbl 1063.44004; Zbl 0976.44001