Negative theorem for LP, 0 < P < 1monotone approximation

Abstract EN

For a given nonnegative integer number n, we can find a monotone function f depending on n, defined on the interval I=[-1,1], and an absolute constant c>0, satisfying the following relationship: ( ́) ( ) ( ) ( ́) , where ( ) is the degree of the best Lp monotone approximation of the function f by algebraic polynomial of degree not exceeding n+1.

( ́) is the degree of the best Lp approximation of the function ́ by algebraic polynomial of degree not exceeding n.