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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.

MSC:
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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