Upper bound for the degree of an approximating monomial (Q912326)

The author investigates the best approximation of polynomials P of degree \(\leq n\) by quasi-monomials \(m(x)=cx^ k\) (i.e. k is real and \(\geq n)\) and proves, for the case of the \(L^ 2\)-norm on the interval [0,1], the following estimate \((n+1)^ 3/4\leq M_ n\leq 6(n+1)^ 3,\) where \(M_ n\) is the best bound \(K_ n\) such that \(K_ n\) are greater than or equal to the orders of the monomials of the best approximation of P \((K_ n\) are independent of P).