### NEGATIVE THEOREM FOR LP,0

#### Abstract

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:

(2〖E_n (f ́ )〗_p)/(n+1)^3 ≤〖E_(n+1)^1 (f)〗_p≤c〖E_n (f ́ )〗_p,

where 〖E_(n+1)^1 (f)〗_p is the degree of the best Lp monotone approximation of the function f by algebraic polynomial of degree not exceeding n+1. 〖E_n (f ́ )〗_p is the degree of the best Lp approximation of the function f ́ by algebraic polynomial of degree not exceeding n.

(2〖E_n (f ́ )〗_p)/(n+1)^3 ≤〖E_(n+1)^1 (f)〗_p≤c〖E_n (f ́ )〗_p,

where 〖E_(n+1)^1 (f)〗_p is the degree of the best Lp monotone approximation of the function f by algebraic polynomial of degree not exceeding n+1. 〖E_n (f ́ )〗_p is the degree of the best Lp approximation of the function f ́ by algebraic polynomial of degree not exceeding n.

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