The Whitney constants are bounded by two (Q1326905)

The following refinement of Whitney's classical result is proved: for any \(f \in C[0;1]\) and an algebraic polynomial of degree at most \((n-1)\) such that \(\int^{i/n}_ 0 (f(t) - P(t))dt = 0\), \(i=1, \dots, n\), the inequality \(\| f - P \|_{C[0;1]} \leq 2 \omega_ n (t)\) hold. (Here \(\omega_ n (f)\) is the \(n\)-th modulo of smoothness for \(f)\).