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We characterize all subsets $M$ of ${ℝ}^n\times {ℝ}^{+}$ such that $$\underset{X\in {ℝ}^n\times {ℝ}^{+}}{sup}u\left(X\right)=\underset{X\in M}{sup}u\left(X\right)$$ for every bounded parabolic function $u$ on ${ℝ}^n\times {ℝ}^{+}$. The closely related problem of representing functions as sums of Weierstrass kernels corresponding to points of $M$ is also considered. The results provide a parabolic counterpart to results for classical harmonic functions in a ball, see References. As a by-product the question of representability of probability continuous distributions as sums of multiples of normal distributions is investigated.

Let $T$ be a positive number or $+\infty$. We characterize all subsets $M$ of ${ℝ}^n\times ]0,T[$ such that $$\underset{X\in {ℝ}^n\times ]0,T[}{inf}u\left(X\right)=\underset{X\in M}{inf}u\left(X\right)i$$ for every positive parabolic function $u$ on ${ℝ}^n\times ]0,T[$ in terms of coparabolic (minimal) thinness of the set $M_{\delta }={\cup }_{(x,t)\in M}{B}^p((x,t),\delta t)$, where $\delta \in (0,1)$ and $B^p((x,t),r)$ is the “heat ball” with the “center” $(x,t)$ and radius $r$. Examples of different types of sets which can be used instead of “heat balls” are given. It is proved that (i) is equivalent to the condition ${sup}_{X\in {ℝ}^n\times {ℝ}^{+}}u\left(X\right)={sup}_{X\in M}u\left(X\right)$ for every bounded parabolic function on ${ℝ}^n\times {ℝ}^{+}$ and hence to all equivalent conditions given in the article [7]....

Let $\alpha >0$, $\lambda ={\left(2\alpha \right)}^{-1/2}$, $S^{n-1}$ be the $(n-1)$-dimensional unit sphere, $\sigma$ be the surface measure on $S^{n-1}$ and $h\left(x\right)={\int }_{S^{n-1}}{e}^{\lambda \langle x,y\rangle }d\sigma \left(y\right)$. We characterize all subsets $M$ of ${ℝ}^n$ such that $$\underset{x\in {ℝ}^n}{inf}\frac{u\left(x\right)}{h\left(x\right)}=\underset{x\in M}{inf}\frac{u\left(x\right)}{h\left(x\right)}$$ for every positive solution $u$ of the Helmholtz equation on ${ℝ}^n$. A closely related problem of representing functions of $L_1\left(S^{n-1}\right)$ as sums of blocks of the form $e^{\lambda \langle x_k,.\rangle }/h\left(x_k\right)$ corresponding to points of $M$ is also considered. The results provide a counterpart to results for classical harmonic functions in a ball, and for parabolic functions on a slab, see References.