Approximate Taylor polynomials and differentiation of functions (Q1333766)

The authors have proved that for a measurable function \(u\) defined on a measurable subset \(D\) of the \(n\)-dimensional Euclidean space the following conditions are equivalent: \(u\) has the Lusin property of order \(k\) on \(D\); \(u\) has an approximate \((k-1)\)-Taylor polynomial at almost every point of \(D\); \(u\) is approximately differentiable of order \(k\) at almost every point of \(D\). Here the Lusin property of order \(k\) means that for each \(\varepsilon> 0\) there exists a \(C^ k\)-function \(g\) defined on \(\mathbb{R}^ n\) such that \(m(\{x\in D: u(x)\neq g(x)\})< \varepsilon\), where \(m\) is a Lebesgue measure, \(p\) is an approximate \((k- 1)\)-Taylor polynomial if \(p(x,y)\) is a polynomial centered at \(x\) of degree at most \((k-1)\) such that \[ \text{ap} \limsup_{y\to x} | u(y)- p(x,y)|\cdot | y- x|^{-k}< +\infty \] and \(u\) is approximately differentiable of order \(k\) if there is a polynomial \(p(x,y)\) centered at \(x\) and of degree at most \(k\) such that \[ \text{ap} \lim_{y\to x} | u(y)- p(x,y)|\cdot| y- x|^{-k}= 0. \] {}.