Harmonic analogues of G. R. MacLane's universal functions. II
[For part I see J. Lond. Math. Soc. (2) 57, No. 1, 148-156 (1998; Zbl 0922.31004).] Let \(H_N\) be the space of all harmonic functions on \(\mathbb{R}^n\) with the topology of local uniform convergence. The authors investigate the question of the existence of a harmonic function \(h\in H_N\), such that \[ \{\partial^{\alpha_1+\cdots+ \alpha_N}/\partial_{x_1}^{\alpha_1}\dots \partial x_N^{\alpha_N} h: (\alpha_1,\dots, \alpha_N)\in J\subset \mathbb{N}^N\} \] is dense in \(H_N\) (a function with this property is called \(J\)-universal function). One of the results obtained is: Theorem 1. If \(J\) is an infinite subset of \(\mathbb{N}^N\), then there exists a \(J\)-universal function of exponential type \(c_N\) where \(c_2=1\) and \[ c_N= N\Biggl( \prod_{j=1}^{N-1} \frac{(2j)^{2j}} {(2j+1)^{2j+1}} \Biggr)^{1/(2N)}, \quad N\geq 3. \] In the case where \(J= \{(m,\dots, m)\in \mathbb{N}^N\}\) there is no \(J\)-universal function of exponential type less than \(\sqrt{N/2}\).
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