Mean value theorems via spectral synthesis (Q1900657)

Let \(v_1, v_2, \dots, v_n\) \((n\geq 2)\) be the vertices of some fixed regular simplex in \(\mathbb{R}^n\) with centre 0 and radius 1. Then the vertices of any regular simplex in \(\mathbb{R}^n\) with centre \(x\) and radius \(r>0\) are \(x+ rU v_0, x+rU v_1, \dots, x+rU v_n\) where \(U\) denotes an appropriately chosen orthogonal \(n\times n\) matrix. The author characterizes those continuous functions \(f: \mathbb{R}^n\to \mathbb{C}\) with the property that, for a fixed \(r>0\), \(\sum^n_{k=0} f(x+ rUv_k)= (n+1) f(x)\) for all \(x\in \mathbb{R}^n\) and all real orthogonal \(n\times n\) matrices \(U\). Indeed \(f\) is precisely a harmonic polynomial of degree at most 2. Another interesting result proved is the following: Suppose \(0\neq \alpha_2\in \mathbb{C}\), \(0\leq \nu\leq N\) (\(N\) is an integer \(\geq 1\)), \(\sum^N_{\nu=0} \alpha_\nu =0\) and \(r_1, r_2, \dots, r_N\) are distinct real numbers. (i) There exists a non-negative integer \(m<N\) such that \(\sum^N_{\nu=0} \alpha_\nu r^j_\nu= 0\) for \(0\leq j\leq m\) and \(\sum^N_{\nu=0} \alpha_\nu r_\nu^{m+1} \neq 0\). (ii) If \(f: \mathbb{R}^n\to \mathbb{C}\), \(n\geq 2\) and \(f\) is continuous then \(\sum^N_{\nu=0} \alpha_\nu f(x+ r_\nu u)=0\) for all \(x\in \mathbb{R}^n\) and all \(u\in S^{n-1}= \{u\in \mathbb{R}^n: |u|=1\}\) iff \(f\) is a polynomial function of degree at most \(m\).