Following the work of \textit{J. Lukeš}, \textit{J. Malý}, \textit{I. Netuka}, \textit{M. Smrcka} and \textit{J. Spurný} [Isr. J. Math. 134, 255--287 (2003; Zbl 1031.35011)], the authors characterize the real-valued functions on a compact set \(K\) in \(\mathbb{R}^n\) that can be expressed as the pointwise limit of a sequence of functions each of which are harmonic on some neighborhood of \(K\). They also characterize the functions on the unit sphere which are the radial limits of entire harmonic functions.