Stability of functional equations connected with quadrature rules (Q264170)

The author proves the stability of the functional equation \[ F(y)-F(x)=(y-x)\sum_{i=1}^{n}a_if(\alpha_ix+(1-\alpha_i)y).\tag{1} \] Namely, for mappings \(f,F:\mathbb{R}\to\mathbb{R}\), \(n\in\mathbb{N}\), \(a_i\in\mathbb{R}\setminus\{0\}\) and distinct \(\alpha_i\in[0,1]\) (\(i=1,\dots,n\)) if \[ \left|\frac{F(y)-F(x)}{y-x}-\sum_{i=1}^{n}a_if\left(\alpha_ix+(1-\alpha_i)y\right)\right|\leq\varepsilon,\qquad x\neq y, \] then there exist polynomial functions \(p\) of order at most \(3n-2\) and \(P\) of order at most \(3n-1\) and constants \(M,K>0\) such that \[ |f(x)-p(x)|<M\varepsilon\quad\text{and}\quad |F(x)-P(x)|<K\varepsilon. \] Moreover, \(p,P\) satisfy Equation (1).