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

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.


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