The Fréchet functional equation with application to the stability of certain operators (Q657439)

The article deals with the Fréchet functional equation \[ \Delta_h^n f(x) = 0, \quad \Delta_h^n f(x) = \sum_{j=0}^n (-1)^j \begin{pmatrix} n \\ j \end{pmatrix} f(x + (n - j)h). \] Solutions of this equation are polynomials of degree \(\leq n\). The main result is the following: if \(f \in C^{k(n)}[0,1]\) satisfies the equation \[ \Delta_h^n f(0) = 0, \quad h \in \bigg[0,\frac1n\bigg], \] where \(k(1) = 0\), \(k(2) = 1\), \(k(3) = 3\), and, for \(n \geq 4\), \[ k(n) = \min \bigg\{k \in {\mathbb N}:\;\sum_{j=1}^{n-1} \begin{pmatrix} n \\ j \end{pmatrix} \bigg(1 - \frac{i}n\bigg)^k < 1\bigg\}, \] then \(f\) is a polynomial of degree at most \(n - 1\). This result is applied to the Hyers-Ulam stability property for the following Bernstein-Schnabl operator \[ B_nf(x) = \int_0^1 \dots \int_0^1 f\bigg(\frac{x_1 + \dots + x_n}{n}\bigg) \, d\mu_x(x_1) \dots d\mu_x(x_n). \]